Geometry in Design
description
Transcript of Geometry in Design
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 160
6Geometry in Design
Why is knowledge of geometry important for so many activities Think
about a concert that you attended How did geometry influence
the band who wrote the music the craftspeople who made the
instruments the artists who decorated the concert hall the
architect who designed the hall the engineers who planned
the hallrsquos construction and the tradespeople whose skills andtalents made it all possible
In this chapter you will
identify real-world applications of geometric shapes andfigures through investigation in a variety of contextsand explain these applications
represent three-dimensional objects using concretematerials and design or drawing software in a varietyof ways
create nets plans and patterns from physical models arising from a variety ofreal-world applications by applying the metric and imperial systems and usingdesign or drawing software
solve design problems that satisfy given constraints using physical models ordrawings and state any assumptions made
Key Terms
constraint
golden ratio
golden rectangle
isometric perspectivedrawing
net
orthographic drawings
orthographic projection
pattern
plan
scale model
tessellation
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
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Arqum studied graphic design at Fanshawe College and now creates
signs for companies selling a product or service He discusses each
project with his customers including size materials budget and
regulations and designs layouts for billboards and business signs
Arqum uses geometric shapes and scale when he creates his designs
on the computer
6 Geometry in Desgin bull MHR
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Geometric Shapes
1 Identiy the two-dimensional geometric
shapes
a) b)
c) d)
2 Identiy the three-dimensional geometric
shapes
a) b)
c) d)
3 Identiy the polygons
a) b)
c) d)
e)
Perimeter and Area
Figure PerimeterCircumference Area
rectangle P = 2l + 2w A = lw
triangle P = a + b + c A = 1
_
2 bh
circle C = πd C = 2πr A = πr 2
4 Find the perimeter and the area o a
rectangular sports 1047297eld that measures
120 m by 100 m
5 Find the perimeter and the area o a
school crest in the shape o an equilateral
triangle with a side length o 15 cm Teheight o the triangle is approximately
13 cm
6 Find the perimeter the and area o the
base o a circular wading pool with a
radius o 12 m
Surface Area and Volume
Figure Surface Area Volume
l w
h
SA = 2lw + 2wh + 2lh V = lwh
h
bl
c
a SA = al + bl + cl + bh V = 1
_
2 blh
h
r
SA = 2πrh + 2πr 2 V = πr 2h
Prerequisite Skills
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Chapter Problem
Paul has secured a co-op position for one semester with
a firm of architects and designers The firm provides
customers with designs drawings scale models
plans patterns and cost estimates for a variety of
projects such as prefabricated outbuildings houses
and industrial products Paul will learn how to provide
customers with the services that the firm offers As he
works through the co-op position Paul will be given
design tasks that will require the knowledge and skills
that he has learned during his placement
7 Wilhelm built a closed room in his garage
to use as a paint shop Te room measured
8 m by 6 m by 25 m a) Wilhelm painted the interior walls and
ceiling Find the area that was painted
b) Wilhelm plans to install a blower to
pump resh air into the room Te size
o blower required depends on the
volume o the room Find the volume o
the room
8 Sunita built a wheelchair ramp in the
shape o a triangular prism as shown
Each surace o the ramp is to be paintedmdash
with the exception o the base Find the
surace area to be painted
09 m
06 m01 m
9 A cylindrical water tank a radius o 5 f
and a height o 12 f Te interior is to bepainted with waterproo paint Find the
area to be painted and the volume o wat
that the tank can hold
Angles in a Polygon
10 Te sum o the angles in a polygon with
n sides in degrees is given by the ormu
S = 180(n - 2) I the polygon is regular
the measure o each angle is given by the
sum divided by n Find the sum o the
angles and the measure o each angle o
a regular pentagon
Scale Factors
11 Anil is planning to grow vegetables in a
market garden measuring 24 m by 36 m
o plan the garden he will draw a diagra
using a scale o 1 cm to represent 2 m
Find the dimensions o the diagram
12 ed has made a scale model o a sailboat
that he is planning to build Te model
has a length o 10 in ed used a scale o1 in to represent 4 f What is the length
the sailboat
Get Ready bull MHR
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What is the link between the cells that make up a cell phone network
and a tesselation pattern How are the Mona Lisa and the Great Pyramid
o Cheops in Egypt related What is the connection between music
and geometry
In this section you will investigate some o these relations and begin to
learn how to use geometric principles in art architecture and ashion
Investigate 1 The Mona Lisa
Tools copy of Leonardo
da Vincirsquos painting
Mona Lisa
ruleror
computer
The Geometerrsquos
Sketchpad reg
Uncover one o the secrets o the Mona Lisa
1 Draw a rectangle that encloses the ace o the
Mona Lisa
Method 1 Use Pencil and Paper
Use your ruler and pencil to draw the rectangle
Measure the length and width o the rectangle
Divide the length by the width
Method 2 Use The Geometerrsquos Sketchpad regTechnology TipIf the picture appears
too large or too small
you can resize it using
any drawing program
such as Microsoftreg
Paint
Open Te Geometerrsquos Sketchpad reg Search the Internet or a picture o
the Mona Lisa Right-click on the picture and select Copy Return to
Te Geometerrsquos Sketchpad reg Select Paste Picture rom the Edit menu
Draw a rectangle around the ace o the Mona Lisa Measure the
length and width o the rectangle Divide the length by the width
Investigate Geometric Shapesand Figures61
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golden ratio
bull the ratio of two lengths
is approximately 16181
golden rectangle
bull a rectangle that is
pleasing to the eye
bull the ratio of the
length to the width isapproximately 16181
2 Te ratio o the length to the width on the actual painting is 16181
Tis is known as the golden ratio A rectangle that is drawn using
the golden ratio is known as a golden rectangle Te proportions
o golden rectangles are pleasing to the eye Te golden ratio is ofen
used in art and architecture
3 Re1047298ect Tink about windows that you have seen on buildings in you
neighbourhood Which windows appear to be golden rectangles
o learn more about the golden ratio in art and architecture go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Investigate 2 Tessellations
Tools centimetre grid paper
coloured pencils or
markers pattern block template
or pattern blocks
tessellation
bull a pattern that covers
a plane without
overlapping or leaving
gaps
bull also called a tiling
pattern
Certain shapes can be used to 1047297ll a plane
completely with no gaps Tis is known as
a tessellation or ldquotiling the planerdquo As you
can see rom a piece o grid paper a square isone o these shapes Clothing is ofen made
rom abrics with a checkerboard pattern I
colours are careully chosen and placed the
pattern becomes more attractive
Method 1 Use Pencil and Paper
1 Draw a square Ten draw additional squares identical to the 1047297rst
square such that they orm a tessellation Use at least our different
colours to make a pleasing pattern or clothing 2 riangles can also be used to orm tessellations Draw a triangle
Ten draw additional triangles identical to the 1047297rst triangle such
that they orm a tessellation Colour the triangles to create a
pleasing pattern
3 Re1047298ect Can you use any triangle to tile the plane ry several example
4 You can use a combination o triangles and rectangles to orm a
tessellation Find a combination o a rectangle and several triangles
that can be used to tile the plane Draw your tessellation Colour yopattern to create a pleasing design
5 You can also use other regular geometric shapes to construct
tessellations Pattern blocks include many different regular geometr
shapes Use the blocks to experiment with different shapes that can
tile the plane Examine a vertex that is common to several polygons
What is the measure o the angle around the vertex
61 Investigate Geometric Shapes and Figures bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
pattern block applet
Method 2 Use Computer Software
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Select both axes the origin and the unit point Choose
Hide Objects rom the Display menu
2 Start in the upper lef corner and draw our points to enclose one
o the squares Select the our points in order From the Construct menu choose Quadrilateral Interior Select the our points and the
interior From the Edit menu choose Copy and then Paste Drag the
image to another square
Technology TipOnce you have a basic
pattern you can select
all elements of the
pattern Then use the
Copy command from
the Edit menu andPaste it repeatedly This
is a fast way of tiling
the plane
3 Right-click on one o the interiors Choose Color and change the
colour o the square You can continue this process to tile the screen
with different coloured squares Use at least our different colours to
design a tessellation that would make a pleasing pattern or clothing
4 riangles can also be used to orm tessellations Start a new sketch
Draw a triangle Ten draw additional triangles identical to the 1047297rsttriangle such that they orm a tessellation Colour the triangles to
orm a pleasing pattern
5 Re1047298ect Can you use any triangle to tile the plane ry several
examples
6 You can use a combination o triangles and rectangles to orm a
tessellation Start with a grid o squares Find a combination o a
rectangle and several triangles that can be used to tile the plane Draw
your tesselation Colour your pattern to orm a pleasing design
7 You can also use other regular geometric shapes to construct
tessellations Go to wwwmcgrawhillcalinksfoundations11 and
ollow the links to the Pattern Block Applet Use the applet to
experiment with different shapes that can tile the plane Examine a
vertex that is common to several polygons What is the measure o
the angle around the vertex
Te only regular polygons (triangle square pentagon and so on) that
can tile the plane are those whose interior angles divide evenly into 360degTese are the triangle (60deg) the square (90deg) and the hexagon (120deg)
A pentagon or example will not tile a plane because its interior angles
(108deg) do not divide evenly into 360deg
o learn more about tessellations go to wwwmcgrawhillcalinks
foundations11 and ollow the links
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Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
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Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
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For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 260
Arqum studied graphic design at Fanshawe College and now creates
signs for companies selling a product or service He discusses each
project with his customers including size materials budget and
regulations and designs layouts for billboards and business signs
Arqum uses geometric shapes and scale when he creates his designs
on the computer
6 Geometry in Desgin bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 360
Geometric Shapes
1 Identiy the two-dimensional geometric
shapes
a) b)
c) d)
2 Identiy the three-dimensional geometric
shapes
a) b)
c) d)
3 Identiy the polygons
a) b)
c) d)
e)
Perimeter and Area
Figure PerimeterCircumference Area
rectangle P = 2l + 2w A = lw
triangle P = a + b + c A = 1
_
2 bh
circle C = πd C = 2πr A = πr 2
4 Find the perimeter and the area o a
rectangular sports 1047297eld that measures
120 m by 100 m
5 Find the perimeter and the area o a
school crest in the shape o an equilateral
triangle with a side length o 15 cm Teheight o the triangle is approximately
13 cm
6 Find the perimeter the and area o the
base o a circular wading pool with a
radius o 12 m
Surface Area and Volume
Figure Surface Area Volume
l w
h
SA = 2lw + 2wh + 2lh V = lwh
h
bl
c
a SA = al + bl + cl + bh V = 1
_
2 blh
h
r
SA = 2πrh + 2πr 2 V = πr 2h
Prerequisite Skills
4 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 460
Chapter Problem
Paul has secured a co-op position for one semester with
a firm of architects and designers The firm provides
customers with designs drawings scale models
plans patterns and cost estimates for a variety of
projects such as prefabricated outbuildings houses
and industrial products Paul will learn how to provide
customers with the services that the firm offers As he
works through the co-op position Paul will be given
design tasks that will require the knowledge and skills
that he has learned during his placement
7 Wilhelm built a closed room in his garage
to use as a paint shop Te room measured
8 m by 6 m by 25 m a) Wilhelm painted the interior walls and
ceiling Find the area that was painted
b) Wilhelm plans to install a blower to
pump resh air into the room Te size
o blower required depends on the
volume o the room Find the volume o
the room
8 Sunita built a wheelchair ramp in the
shape o a triangular prism as shown
Each surace o the ramp is to be paintedmdash
with the exception o the base Find the
surace area to be painted
09 m
06 m01 m
9 A cylindrical water tank a radius o 5 f
and a height o 12 f Te interior is to bepainted with waterproo paint Find the
area to be painted and the volume o wat
that the tank can hold
Angles in a Polygon
10 Te sum o the angles in a polygon with
n sides in degrees is given by the ormu
S = 180(n - 2) I the polygon is regular
the measure o each angle is given by the
sum divided by n Find the sum o the
angles and the measure o each angle o
a regular pentagon
Scale Factors
11 Anil is planning to grow vegetables in a
market garden measuring 24 m by 36 m
o plan the garden he will draw a diagra
using a scale o 1 cm to represent 2 m
Find the dimensions o the diagram
12 ed has made a scale model o a sailboat
that he is planning to build Te model
has a length o 10 in ed used a scale o1 in to represent 4 f What is the length
the sailboat
Get Ready bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 560
What is the link between the cells that make up a cell phone network
and a tesselation pattern How are the Mona Lisa and the Great Pyramid
o Cheops in Egypt related What is the connection between music
and geometry
In this section you will investigate some o these relations and begin to
learn how to use geometric principles in art architecture and ashion
Investigate 1 The Mona Lisa
Tools copy of Leonardo
da Vincirsquos painting
Mona Lisa
ruleror
computer
The Geometerrsquos
Sketchpad reg
Uncover one o the secrets o the Mona Lisa
1 Draw a rectangle that encloses the ace o the
Mona Lisa
Method 1 Use Pencil and Paper
Use your ruler and pencil to draw the rectangle
Measure the length and width o the rectangle
Divide the length by the width
Method 2 Use The Geometerrsquos Sketchpad regTechnology TipIf the picture appears
too large or too small
you can resize it using
any drawing program
such as Microsoftreg
Paint
Open Te Geometerrsquos Sketchpad reg Search the Internet or a picture o
the Mona Lisa Right-click on the picture and select Copy Return to
Te Geometerrsquos Sketchpad reg Select Paste Picture rom the Edit menu
Draw a rectangle around the ace o the Mona Lisa Measure the
length and width o the rectangle Divide the length by the width
Investigate Geometric Shapesand Figures61
6 MHR bull Chapter 8
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 660
golden ratio
bull the ratio of two lengths
is approximately 16181
golden rectangle
bull a rectangle that is
pleasing to the eye
bull the ratio of the
length to the width isapproximately 16181
2 Te ratio o the length to the width on the actual painting is 16181
Tis is known as the golden ratio A rectangle that is drawn using
the golden ratio is known as a golden rectangle Te proportions
o golden rectangles are pleasing to the eye Te golden ratio is ofen
used in art and architecture
3 Re1047298ect Tink about windows that you have seen on buildings in you
neighbourhood Which windows appear to be golden rectangles
o learn more about the golden ratio in art and architecture go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Investigate 2 Tessellations
Tools centimetre grid paper
coloured pencils or
markers pattern block template
or pattern blocks
tessellation
bull a pattern that covers
a plane without
overlapping or leaving
gaps
bull also called a tiling
pattern
Certain shapes can be used to 1047297ll a plane
completely with no gaps Tis is known as
a tessellation or ldquotiling the planerdquo As you
can see rom a piece o grid paper a square isone o these shapes Clothing is ofen made
rom abrics with a checkerboard pattern I
colours are careully chosen and placed the
pattern becomes more attractive
Method 1 Use Pencil and Paper
1 Draw a square Ten draw additional squares identical to the 1047297rst
square such that they orm a tessellation Use at least our different
colours to make a pleasing pattern or clothing 2 riangles can also be used to orm tessellations Draw a triangle
Ten draw additional triangles identical to the 1047297rst triangle such
that they orm a tessellation Colour the triangles to create a
pleasing pattern
3 Re1047298ect Can you use any triangle to tile the plane ry several example
4 You can use a combination o triangles and rectangles to orm a
tessellation Find a combination o a rectangle and several triangles
that can be used to tile the plane Draw your tessellation Colour yopattern to create a pleasing design
5 You can also use other regular geometric shapes to construct
tessellations Pattern blocks include many different regular geometr
shapes Use the blocks to experiment with different shapes that can
tile the plane Examine a vertex that is common to several polygons
What is the measure o the angle around the vertex
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
pattern block applet
Method 2 Use Computer Software
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Select both axes the origin and the unit point Choose
Hide Objects rom the Display menu
2 Start in the upper lef corner and draw our points to enclose one
o the squares Select the our points in order From the Construct menu choose Quadrilateral Interior Select the our points and the
interior From the Edit menu choose Copy and then Paste Drag the
image to another square
Technology TipOnce you have a basic
pattern you can select
all elements of the
pattern Then use the
Copy command from
the Edit menu andPaste it repeatedly This
is a fast way of tiling
the plane
3 Right-click on one o the interiors Choose Color and change the
colour o the square You can continue this process to tile the screen
with different coloured squares Use at least our different colours to
design a tessellation that would make a pleasing pattern or clothing
4 riangles can also be used to orm tessellations Start a new sketch
Draw a triangle Ten draw additional triangles identical to the 1047297rsttriangle such that they orm a tessellation Colour the triangles to
orm a pleasing pattern
5 Re1047298ect Can you use any triangle to tile the plane ry several
examples
6 You can use a combination o triangles and rectangles to orm a
tessellation Start with a grid o squares Find a combination o a
rectangle and several triangles that can be used to tile the plane Draw
your tesselation Colour your pattern to orm a pleasing design
7 You can also use other regular geometric shapes to construct
tessellations Go to wwwmcgrawhillcalinksfoundations11 and
ollow the links to the Pattern Block Applet Use the applet to
experiment with different shapes that can tile the plane Examine a
vertex that is common to several polygons What is the measure o
the angle around the vertex
Te only regular polygons (triangle square pentagon and so on) that
can tile the plane are those whose interior angles divide evenly into 360degTese are the triangle (60deg) the square (90deg) and the hexagon (120deg)
A pentagon or example will not tile a plane because its interior angles
(108deg) do not divide evenly into 360deg
o learn more about tessellations go to wwwmcgrawhillcalinks
foundations11 and ollow the links
8 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
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For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 360
Geometric Shapes
1 Identiy the two-dimensional geometric
shapes
a) b)
c) d)
2 Identiy the three-dimensional geometric
shapes
a) b)
c) d)
3 Identiy the polygons
a) b)
c) d)
e)
Perimeter and Area
Figure PerimeterCircumference Area
rectangle P = 2l + 2w A = lw
triangle P = a + b + c A = 1
_
2 bh
circle C = πd C = 2πr A = πr 2
4 Find the perimeter and the area o a
rectangular sports 1047297eld that measures
120 m by 100 m
5 Find the perimeter and the area o a
school crest in the shape o an equilateral
triangle with a side length o 15 cm Teheight o the triangle is approximately
13 cm
6 Find the perimeter the and area o the
base o a circular wading pool with a
radius o 12 m
Surface Area and Volume
Figure Surface Area Volume
l w
h
SA = 2lw + 2wh + 2lh V = lwh
h
bl
c
a SA = al + bl + cl + bh V = 1
_
2 blh
h
r
SA = 2πrh + 2πr 2 V = πr 2h
Prerequisite Skills
4 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 460
Chapter Problem
Paul has secured a co-op position for one semester with
a firm of architects and designers The firm provides
customers with designs drawings scale models
plans patterns and cost estimates for a variety of
projects such as prefabricated outbuildings houses
and industrial products Paul will learn how to provide
customers with the services that the firm offers As he
works through the co-op position Paul will be given
design tasks that will require the knowledge and skills
that he has learned during his placement
7 Wilhelm built a closed room in his garage
to use as a paint shop Te room measured
8 m by 6 m by 25 m a) Wilhelm painted the interior walls and
ceiling Find the area that was painted
b) Wilhelm plans to install a blower to
pump resh air into the room Te size
o blower required depends on the
volume o the room Find the volume o
the room
8 Sunita built a wheelchair ramp in the
shape o a triangular prism as shown
Each surace o the ramp is to be paintedmdash
with the exception o the base Find the
surace area to be painted
09 m
06 m01 m
9 A cylindrical water tank a radius o 5 f
and a height o 12 f Te interior is to bepainted with waterproo paint Find the
area to be painted and the volume o wat
that the tank can hold
Angles in a Polygon
10 Te sum o the angles in a polygon with
n sides in degrees is given by the ormu
S = 180(n - 2) I the polygon is regular
the measure o each angle is given by the
sum divided by n Find the sum o the
angles and the measure o each angle o
a regular pentagon
Scale Factors
11 Anil is planning to grow vegetables in a
market garden measuring 24 m by 36 m
o plan the garden he will draw a diagra
using a scale o 1 cm to represent 2 m
Find the dimensions o the diagram
12 ed has made a scale model o a sailboat
that he is planning to build Te model
has a length o 10 in ed used a scale o1 in to represent 4 f What is the length
the sailboat
Get Ready bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 560
What is the link between the cells that make up a cell phone network
and a tesselation pattern How are the Mona Lisa and the Great Pyramid
o Cheops in Egypt related What is the connection between music
and geometry
In this section you will investigate some o these relations and begin to
learn how to use geometric principles in art architecture and ashion
Investigate 1 The Mona Lisa
Tools copy of Leonardo
da Vincirsquos painting
Mona Lisa
ruleror
computer
The Geometerrsquos
Sketchpad reg
Uncover one o the secrets o the Mona Lisa
1 Draw a rectangle that encloses the ace o the
Mona Lisa
Method 1 Use Pencil and Paper
Use your ruler and pencil to draw the rectangle
Measure the length and width o the rectangle
Divide the length by the width
Method 2 Use The Geometerrsquos Sketchpad regTechnology TipIf the picture appears
too large or too small
you can resize it using
any drawing program
such as Microsoftreg
Paint
Open Te Geometerrsquos Sketchpad reg Search the Internet or a picture o
the Mona Lisa Right-click on the picture and select Copy Return to
Te Geometerrsquos Sketchpad reg Select Paste Picture rom the Edit menu
Draw a rectangle around the ace o the Mona Lisa Measure the
length and width o the rectangle Divide the length by the width
Investigate Geometric Shapesand Figures61
6 MHR bull Chapter 8
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 660
golden ratio
bull the ratio of two lengths
is approximately 16181
golden rectangle
bull a rectangle that is
pleasing to the eye
bull the ratio of the
length to the width isapproximately 16181
2 Te ratio o the length to the width on the actual painting is 16181
Tis is known as the golden ratio A rectangle that is drawn using
the golden ratio is known as a golden rectangle Te proportions
o golden rectangles are pleasing to the eye Te golden ratio is ofen
used in art and architecture
3 Re1047298ect Tink about windows that you have seen on buildings in you
neighbourhood Which windows appear to be golden rectangles
o learn more about the golden ratio in art and architecture go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Investigate 2 Tessellations
Tools centimetre grid paper
coloured pencils or
markers pattern block template
or pattern blocks
tessellation
bull a pattern that covers
a plane without
overlapping or leaving
gaps
bull also called a tiling
pattern
Certain shapes can be used to 1047297ll a plane
completely with no gaps Tis is known as
a tessellation or ldquotiling the planerdquo As you
can see rom a piece o grid paper a square isone o these shapes Clothing is ofen made
rom abrics with a checkerboard pattern I
colours are careully chosen and placed the
pattern becomes more attractive
Method 1 Use Pencil and Paper
1 Draw a square Ten draw additional squares identical to the 1047297rst
square such that they orm a tessellation Use at least our different
colours to make a pleasing pattern or clothing 2 riangles can also be used to orm tessellations Draw a triangle
Ten draw additional triangles identical to the 1047297rst triangle such
that they orm a tessellation Colour the triangles to create a
pleasing pattern
3 Re1047298ect Can you use any triangle to tile the plane ry several example
4 You can use a combination o triangles and rectangles to orm a
tessellation Find a combination o a rectangle and several triangles
that can be used to tile the plane Draw your tessellation Colour yopattern to create a pleasing design
5 You can also use other regular geometric shapes to construct
tessellations Pattern blocks include many different regular geometr
shapes Use the blocks to experiment with different shapes that can
tile the plane Examine a vertex that is common to several polygons
What is the measure o the angle around the vertex
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 760
Tools computer
The Geometerrsquos
Sketchpad reg
pattern block applet
Method 2 Use Computer Software
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Select both axes the origin and the unit point Choose
Hide Objects rom the Display menu
2 Start in the upper lef corner and draw our points to enclose one
o the squares Select the our points in order From the Construct menu choose Quadrilateral Interior Select the our points and the
interior From the Edit menu choose Copy and then Paste Drag the
image to another square
Technology TipOnce you have a basic
pattern you can select
all elements of the
pattern Then use the
Copy command from
the Edit menu andPaste it repeatedly This
is a fast way of tiling
the plane
3 Right-click on one o the interiors Choose Color and change the
colour o the square You can continue this process to tile the screen
with different coloured squares Use at least our different colours to
design a tessellation that would make a pleasing pattern or clothing
4 riangles can also be used to orm tessellations Start a new sketch
Draw a triangle Ten draw additional triangles identical to the 1047297rsttriangle such that they orm a tessellation Colour the triangles to
orm a pleasing pattern
5 Re1047298ect Can you use any triangle to tile the plane ry several
examples
6 You can use a combination o triangles and rectangles to orm a
tessellation Start with a grid o squares Find a combination o a
rectangle and several triangles that can be used to tile the plane Draw
your tesselation Colour your pattern to orm a pleasing design
7 You can also use other regular geometric shapes to construct
tessellations Go to wwwmcgrawhillcalinksfoundations11 and
ollow the links to the Pattern Block Applet Use the applet to
experiment with different shapes that can tile the plane Examine a
vertex that is common to several polygons What is the measure o
the angle around the vertex
Te only regular polygons (triangle square pentagon and so on) that
can tile the plane are those whose interior angles divide evenly into 360degTese are the triangle (60deg) the square (90deg) and the hexagon (120deg)
A pentagon or example will not tile a plane because its interior angles
(108deg) do not divide evenly into 360deg
o learn more about tessellations go to wwwmcgrawhillcalinks
foundations11 and ollow the links
8 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 860
Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 960
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1060
Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1260
For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1460
Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
7212019 Geometry in Design
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 460
Chapter Problem
Paul has secured a co-op position for one semester with
a firm of architects and designers The firm provides
customers with designs drawings scale models
plans patterns and cost estimates for a variety of
projects such as prefabricated outbuildings houses
and industrial products Paul will learn how to provide
customers with the services that the firm offers As he
works through the co-op position Paul will be given
design tasks that will require the knowledge and skills
that he has learned during his placement
7 Wilhelm built a closed room in his garage
to use as a paint shop Te room measured
8 m by 6 m by 25 m a) Wilhelm painted the interior walls and
ceiling Find the area that was painted
b) Wilhelm plans to install a blower to
pump resh air into the room Te size
o blower required depends on the
volume o the room Find the volume o
the room
8 Sunita built a wheelchair ramp in the
shape o a triangular prism as shown
Each surace o the ramp is to be paintedmdash
with the exception o the base Find the
surace area to be painted
09 m
06 m01 m
9 A cylindrical water tank a radius o 5 f
and a height o 12 f Te interior is to bepainted with waterproo paint Find the
area to be painted and the volume o wat
that the tank can hold
Angles in a Polygon
10 Te sum o the angles in a polygon with
n sides in degrees is given by the ormu
S = 180(n - 2) I the polygon is regular
the measure o each angle is given by the
sum divided by n Find the sum o the
angles and the measure o each angle o
a regular pentagon
Scale Factors
11 Anil is planning to grow vegetables in a
market garden measuring 24 m by 36 m
o plan the garden he will draw a diagra
using a scale o 1 cm to represent 2 m
Find the dimensions o the diagram
12 ed has made a scale model o a sailboat
that he is planning to build Te model
has a length o 10 in ed used a scale o1 in to represent 4 f What is the length
the sailboat
Get Ready bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 560
What is the link between the cells that make up a cell phone network
and a tesselation pattern How are the Mona Lisa and the Great Pyramid
o Cheops in Egypt related What is the connection between music
and geometry
In this section you will investigate some o these relations and begin to
learn how to use geometric principles in art architecture and ashion
Investigate 1 The Mona Lisa
Tools copy of Leonardo
da Vincirsquos painting
Mona Lisa
ruleror
computer
The Geometerrsquos
Sketchpad reg
Uncover one o the secrets o the Mona Lisa
1 Draw a rectangle that encloses the ace o the
Mona Lisa
Method 1 Use Pencil and Paper
Use your ruler and pencil to draw the rectangle
Measure the length and width o the rectangle
Divide the length by the width
Method 2 Use The Geometerrsquos Sketchpad regTechnology TipIf the picture appears
too large or too small
you can resize it using
any drawing program
such as Microsoftreg
Paint
Open Te Geometerrsquos Sketchpad reg Search the Internet or a picture o
the Mona Lisa Right-click on the picture and select Copy Return to
Te Geometerrsquos Sketchpad reg Select Paste Picture rom the Edit menu
Draw a rectangle around the ace o the Mona Lisa Measure the
length and width o the rectangle Divide the length by the width
Investigate Geometric Shapesand Figures61
6 MHR bull Chapter 8
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 660
golden ratio
bull the ratio of two lengths
is approximately 16181
golden rectangle
bull a rectangle that is
pleasing to the eye
bull the ratio of the
length to the width isapproximately 16181
2 Te ratio o the length to the width on the actual painting is 16181
Tis is known as the golden ratio A rectangle that is drawn using
the golden ratio is known as a golden rectangle Te proportions
o golden rectangles are pleasing to the eye Te golden ratio is ofen
used in art and architecture
3 Re1047298ect Tink about windows that you have seen on buildings in you
neighbourhood Which windows appear to be golden rectangles
o learn more about the golden ratio in art and architecture go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Investigate 2 Tessellations
Tools centimetre grid paper
coloured pencils or
markers pattern block template
or pattern blocks
tessellation
bull a pattern that covers
a plane without
overlapping or leaving
gaps
bull also called a tiling
pattern
Certain shapes can be used to 1047297ll a plane
completely with no gaps Tis is known as
a tessellation or ldquotiling the planerdquo As you
can see rom a piece o grid paper a square isone o these shapes Clothing is ofen made
rom abrics with a checkerboard pattern I
colours are careully chosen and placed the
pattern becomes more attractive
Method 1 Use Pencil and Paper
1 Draw a square Ten draw additional squares identical to the 1047297rst
square such that they orm a tessellation Use at least our different
colours to make a pleasing pattern or clothing 2 riangles can also be used to orm tessellations Draw a triangle
Ten draw additional triangles identical to the 1047297rst triangle such
that they orm a tessellation Colour the triangles to create a
pleasing pattern
3 Re1047298ect Can you use any triangle to tile the plane ry several example
4 You can use a combination o triangles and rectangles to orm a
tessellation Find a combination o a rectangle and several triangles
that can be used to tile the plane Draw your tessellation Colour yopattern to create a pleasing design
5 You can also use other regular geometric shapes to construct
tessellations Pattern blocks include many different regular geometr
shapes Use the blocks to experiment with different shapes that can
tile the plane Examine a vertex that is common to several polygons
What is the measure o the angle around the vertex
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
pattern block applet
Method 2 Use Computer Software
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Select both axes the origin and the unit point Choose
Hide Objects rom the Display menu
2 Start in the upper lef corner and draw our points to enclose one
o the squares Select the our points in order From the Construct menu choose Quadrilateral Interior Select the our points and the
interior From the Edit menu choose Copy and then Paste Drag the
image to another square
Technology TipOnce you have a basic
pattern you can select
all elements of the
pattern Then use the
Copy command from
the Edit menu andPaste it repeatedly This
is a fast way of tiling
the plane
3 Right-click on one o the interiors Choose Color and change the
colour o the square You can continue this process to tile the screen
with different coloured squares Use at least our different colours to
design a tessellation that would make a pleasing pattern or clothing
4 riangles can also be used to orm tessellations Start a new sketch
Draw a triangle Ten draw additional triangles identical to the 1047297rsttriangle such that they orm a tessellation Colour the triangles to
orm a pleasing pattern
5 Re1047298ect Can you use any triangle to tile the plane ry several
examples
6 You can use a combination o triangles and rectangles to orm a
tessellation Start with a grid o squares Find a combination o a
rectangle and several triangles that can be used to tile the plane Draw
your tesselation Colour your pattern to orm a pleasing design
7 You can also use other regular geometric shapes to construct
tessellations Go to wwwmcgrawhillcalinksfoundations11 and
ollow the links to the Pattern Block Applet Use the applet to
experiment with different shapes that can tile the plane Examine a
vertex that is common to several polygons What is the measure o
the angle around the vertex
Te only regular polygons (triangle square pentagon and so on) that
can tile the plane are those whose interior angles divide evenly into 360degTese are the triangle (60deg) the square (90deg) and the hexagon (120deg)
A pentagon or example will not tile a plane because its interior angles
(108deg) do not divide evenly into 360deg
o learn more about tessellations go to wwwmcgrawhillcalinks
foundations11 and ollow the links
8 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 860
Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1060
Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1260
For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1460
Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1660
Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1860
Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1960
Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2060
c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2460
Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
7212019 Geometry in Design
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2860
The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2960
Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3360
9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3460
Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 560
What is the link between the cells that make up a cell phone network
and a tesselation pattern How are the Mona Lisa and the Great Pyramid
o Cheops in Egypt related What is the connection between music
and geometry
In this section you will investigate some o these relations and begin to
learn how to use geometric principles in art architecture and ashion
Investigate 1 The Mona Lisa
Tools copy of Leonardo
da Vincirsquos painting
Mona Lisa
ruleror
computer
The Geometerrsquos
Sketchpad reg
Uncover one o the secrets o the Mona Lisa
1 Draw a rectangle that encloses the ace o the
Mona Lisa
Method 1 Use Pencil and Paper
Use your ruler and pencil to draw the rectangle
Measure the length and width o the rectangle
Divide the length by the width
Method 2 Use The Geometerrsquos Sketchpad regTechnology TipIf the picture appears
too large or too small
you can resize it using
any drawing program
such as Microsoftreg
Paint
Open Te Geometerrsquos Sketchpad reg Search the Internet or a picture o
the Mona Lisa Right-click on the picture and select Copy Return to
Te Geometerrsquos Sketchpad reg Select Paste Picture rom the Edit menu
Draw a rectangle around the ace o the Mona Lisa Measure the
length and width o the rectangle Divide the length by the width
Investigate Geometric Shapesand Figures61
6 MHR bull Chapter 8
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 660
golden ratio
bull the ratio of two lengths
is approximately 16181
golden rectangle
bull a rectangle that is
pleasing to the eye
bull the ratio of the
length to the width isapproximately 16181
2 Te ratio o the length to the width on the actual painting is 16181
Tis is known as the golden ratio A rectangle that is drawn using
the golden ratio is known as a golden rectangle Te proportions
o golden rectangles are pleasing to the eye Te golden ratio is ofen
used in art and architecture
3 Re1047298ect Tink about windows that you have seen on buildings in you
neighbourhood Which windows appear to be golden rectangles
o learn more about the golden ratio in art and architecture go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Investigate 2 Tessellations
Tools centimetre grid paper
coloured pencils or
markers pattern block template
or pattern blocks
tessellation
bull a pattern that covers
a plane without
overlapping or leaving
gaps
bull also called a tiling
pattern
Certain shapes can be used to 1047297ll a plane
completely with no gaps Tis is known as
a tessellation or ldquotiling the planerdquo As you
can see rom a piece o grid paper a square isone o these shapes Clothing is ofen made
rom abrics with a checkerboard pattern I
colours are careully chosen and placed the
pattern becomes more attractive
Method 1 Use Pencil and Paper
1 Draw a square Ten draw additional squares identical to the 1047297rst
square such that they orm a tessellation Use at least our different
colours to make a pleasing pattern or clothing 2 riangles can also be used to orm tessellations Draw a triangle
Ten draw additional triangles identical to the 1047297rst triangle such
that they orm a tessellation Colour the triangles to create a
pleasing pattern
3 Re1047298ect Can you use any triangle to tile the plane ry several example
4 You can use a combination o triangles and rectangles to orm a
tessellation Find a combination o a rectangle and several triangles
that can be used to tile the plane Draw your tessellation Colour yopattern to create a pleasing design
5 You can also use other regular geometric shapes to construct
tessellations Pattern blocks include many different regular geometr
shapes Use the blocks to experiment with different shapes that can
tile the plane Examine a vertex that is common to several polygons
What is the measure o the angle around the vertex
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
pattern block applet
Method 2 Use Computer Software
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Select both axes the origin and the unit point Choose
Hide Objects rom the Display menu
2 Start in the upper lef corner and draw our points to enclose one
o the squares Select the our points in order From the Construct menu choose Quadrilateral Interior Select the our points and the
interior From the Edit menu choose Copy and then Paste Drag the
image to another square
Technology TipOnce you have a basic
pattern you can select
all elements of the
pattern Then use the
Copy command from
the Edit menu andPaste it repeatedly This
is a fast way of tiling
the plane
3 Right-click on one o the interiors Choose Color and change the
colour o the square You can continue this process to tile the screen
with different coloured squares Use at least our different colours to
design a tessellation that would make a pleasing pattern or clothing
4 riangles can also be used to orm tessellations Start a new sketch
Draw a triangle Ten draw additional triangles identical to the 1047297rsttriangle such that they orm a tessellation Colour the triangles to
orm a pleasing pattern
5 Re1047298ect Can you use any triangle to tile the plane ry several
examples
6 You can use a combination o triangles and rectangles to orm a
tessellation Start with a grid o squares Find a combination o a
rectangle and several triangles that can be used to tile the plane Draw
your tesselation Colour your pattern to orm a pleasing design
7 You can also use other regular geometric shapes to construct
tessellations Go to wwwmcgrawhillcalinksfoundations11 and
ollow the links to the Pattern Block Applet Use the applet to
experiment with different shapes that can tile the plane Examine a
vertex that is common to several polygons What is the measure o
the angle around the vertex
Te only regular polygons (triangle square pentagon and so on) that
can tile the plane are those whose interior angles divide evenly into 360degTese are the triangle (60deg) the square (90deg) and the hexagon (120deg)
A pentagon or example will not tile a plane because its interior angles
(108deg) do not divide evenly into 360deg
o learn more about tessellations go to wwwmcgrawhillcalinks
foundations11 and ollow the links
8 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 860
Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
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For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 660
golden ratio
bull the ratio of two lengths
is approximately 16181
golden rectangle
bull a rectangle that is
pleasing to the eye
bull the ratio of the
length to the width isapproximately 16181
2 Te ratio o the length to the width on the actual painting is 16181
Tis is known as the golden ratio A rectangle that is drawn using
the golden ratio is known as a golden rectangle Te proportions
o golden rectangles are pleasing to the eye Te golden ratio is ofen
used in art and architecture
3 Re1047298ect Tink about windows that you have seen on buildings in you
neighbourhood Which windows appear to be golden rectangles
o learn more about the golden ratio in art and architecture go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Investigate 2 Tessellations
Tools centimetre grid paper
coloured pencils or
markers pattern block template
or pattern blocks
tessellation
bull a pattern that covers
a plane without
overlapping or leaving
gaps
bull also called a tiling
pattern
Certain shapes can be used to 1047297ll a plane
completely with no gaps Tis is known as
a tessellation or ldquotiling the planerdquo As you
can see rom a piece o grid paper a square isone o these shapes Clothing is ofen made
rom abrics with a checkerboard pattern I
colours are careully chosen and placed the
pattern becomes more attractive
Method 1 Use Pencil and Paper
1 Draw a square Ten draw additional squares identical to the 1047297rst
square such that they orm a tessellation Use at least our different
colours to make a pleasing pattern or clothing 2 riangles can also be used to orm tessellations Draw a triangle
Ten draw additional triangles identical to the 1047297rst triangle such
that they orm a tessellation Colour the triangles to create a
pleasing pattern
3 Re1047298ect Can you use any triangle to tile the plane ry several example
4 You can use a combination o triangles and rectangles to orm a
tessellation Find a combination o a rectangle and several triangles
that can be used to tile the plane Draw your tessellation Colour yopattern to create a pleasing design
5 You can also use other regular geometric shapes to construct
tessellations Pattern blocks include many different regular geometr
shapes Use the blocks to experiment with different shapes that can
tile the plane Examine a vertex that is common to several polygons
What is the measure o the angle around the vertex
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
pattern block applet
Method 2 Use Computer Software
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Select both axes the origin and the unit point Choose
Hide Objects rom the Display menu
2 Start in the upper lef corner and draw our points to enclose one
o the squares Select the our points in order From the Construct menu choose Quadrilateral Interior Select the our points and the
interior From the Edit menu choose Copy and then Paste Drag the
image to another square
Technology TipOnce you have a basic
pattern you can select
all elements of the
pattern Then use the
Copy command from
the Edit menu andPaste it repeatedly This
is a fast way of tiling
the plane
3 Right-click on one o the interiors Choose Color and change the
colour o the square You can continue this process to tile the screen
with different coloured squares Use at least our different colours to
design a tessellation that would make a pleasing pattern or clothing
4 riangles can also be used to orm tessellations Start a new sketch
Draw a triangle Ten draw additional triangles identical to the 1047297rsttriangle such that they orm a tessellation Colour the triangles to
orm a pleasing pattern
5 Re1047298ect Can you use any triangle to tile the plane ry several
examples
6 You can use a combination o triangles and rectangles to orm a
tessellation Start with a grid o squares Find a combination o a
rectangle and several triangles that can be used to tile the plane Draw
your tesselation Colour your pattern to orm a pleasing design
7 You can also use other regular geometric shapes to construct
tessellations Go to wwwmcgrawhillcalinksfoundations11 and
ollow the links to the Pattern Block Applet Use the applet to
experiment with different shapes that can tile the plane Examine a
vertex that is common to several polygons What is the measure o
the angle around the vertex
Te only regular polygons (triangle square pentagon and so on) that
can tile the plane are those whose interior angles divide evenly into 360degTese are the triangle (60deg) the square (90deg) and the hexagon (120deg)
A pentagon or example will not tile a plane because its interior angles
(108deg) do not divide evenly into 360deg
o learn more about tessellations go to wwwmcgrawhillcalinks
foundations11 and ollow the links
8 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 860
Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
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For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 760
Tools computer
The Geometerrsquos
Sketchpad reg
pattern block applet
Method 2 Use Computer Software
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Select both axes the origin and the unit point Choose
Hide Objects rom the Display menu
2 Start in the upper lef corner and draw our points to enclose one
o the squares Select the our points in order From the Construct menu choose Quadrilateral Interior Select the our points and the
interior From the Edit menu choose Copy and then Paste Drag the
image to another square
Technology TipOnce you have a basic
pattern you can select
all elements of the
pattern Then use the
Copy command from
the Edit menu andPaste it repeatedly This
is a fast way of tiling
the plane
3 Right-click on one o the interiors Choose Color and change the
colour o the square You can continue this process to tile the screen
with different coloured squares Use at least our different colours to
design a tessellation that would make a pleasing pattern or clothing
4 riangles can also be used to orm tessellations Start a new sketch
Draw a triangle Ten draw additional triangles identical to the 1047297rsttriangle such that they orm a tessellation Colour the triangles to
orm a pleasing pattern
5 Re1047298ect Can you use any triangle to tile the plane ry several
examples
6 You can use a combination o triangles and rectangles to orm a
tessellation Start with a grid o squares Find a combination o a
rectangle and several triangles that can be used to tile the plane Draw
your tesselation Colour your pattern to orm a pleasing design
7 You can also use other regular geometric shapes to construct
tessellations Go to wwwmcgrawhillcalinksfoundations11 and
ollow the links to the Pattern Block Applet Use the applet to
experiment with different shapes that can tile the plane Examine a
vertex that is common to several polygons What is the measure o
the angle around the vertex
Te only regular polygons (triangle square pentagon and so on) that
can tile the plane are those whose interior angles divide evenly into 360degTese are the triangle (60deg) the square (90deg) and the hexagon (120deg)
A pentagon or example will not tile a plane because its interior angles
(108deg) do not divide evenly into 360deg
o learn more about tessellations go to wwwmcgrawhillcalinks
foundations11 and ollow the links
8 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 860
Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 960
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1060
Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1160
Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1260
For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1460
Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1660
Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3460
Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3860
Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 860
Investigate 3 Cell Phone Tower Networks
Cell phones are really small radios that
communicate with towers connected to
the telephone system Te towers in an
urban area must be placed close together
because cell phones are low-poweredtypically about 3 W and have a limited
range Also each tower only offers a
limited number o requencies (about
800) restricting the number o calls that
a tower can handle at one time
In an urban area towers are placed about 5 km apart Each tower is
thought o as the centre o a hexagonal cell as shown Consider an urba
area that measures 20 km on a side Draw a tessellation using hexagons
and show how many towers are needed to service the cell phone networ
Tools ruler
grid paper
Method 1 Use Pencil and Paper
1 Let 1 cm represent 1 km Draw a square to represent an area 20 km 20 km
2 Lightly draw a hexagon with a dot in the centre to represent the
tower Draw another hexagon next to the 1047297rst one leaving no gaps
Measure the distance between the two towers I it is not close to
5 cm adjust your hexagons Once you think you have the correct siz
o hexagon continue your diagram to cover the entire area
3 How many towers are required to service the urban area
4 Re1047298ect What are the similarities between tiling the plane withtriangles designing a pattern or clothing and ensuring that cell
phone users have good service
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
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For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1860
Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2060
c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 960
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg From the Graph menu choose
Show Grid Drag the unit point on the horizontal axis until you can
see +10 and -10 on your screen on both the horizontal and vertical
axes Use the Segment ool to draw a rectangle measuring 20 squares
by 20 squares Tis will represent an area 20 km by 20 km
Technology TipDo not worry if your
hexagons look a bit
irregular You will learn
how to draw regular
geometric shapes
accurately later in this
chapter
2 Use the Segment ool to construct a hexagon in the upper lef hand
corner o the 20 km by 20 km grid Draw a point in the centre o your
hexagon to represent the tower
3 Draw another hexagon next to the 1047297rst one leaving no gaps Determine
the distance between the two towers I it is not close to 5 km adjust
your hexagons by dragging vertices Once you think you have the
correct size o hexagon continue your diagram to cover the entire area
4 How many towers are required to service the urban area
5 Re1047298ect What are the similarities between tiling the plane with
triangles designing a pattern or clothing and ensuring that cell
phone users have good service
o learn more about cell phones and how they work go to
wwwmcgrawhillcalinksfoundations11 and ollow the links
Example The Golden Ratio in Ancient Architecture
A cross-section o the Great Pyramid o Cheops on the Giza plateau in
Egypt is shown Find the ratio o the slant height s to hal o the base
Express your answer to three decimal places
s
146515 m
230363 m
10 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1060
Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1160
Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1260
For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3860
Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1060
Solution
Hal o the base = 230363 divide 2
115182
Use the Pythagorean theorem to 1047297nd the slant height
s2 (115182)2 + (146515)2
s2 34 733538 s radic
_________
34 733538
s 186369
Te slant height is approximately 186369 m
Slant heighthal o the base 186369115182
16181
The number given for
the golden ratio 1618
is an approximation
The exact value is
1 + radic __
5_
2 Use a
calculator to verify
that this expression
gives the correct
approximation
Math Connect
Te ratio in the Example is the golden ratio once again Te golden ratio
is a connection between the Mona Lisa and the Great Pyramid A triang
that has this property such as the cross-section o the Pyramid o Cheop
is known as an Egyptian triangle
o learn more about the Pyramids go to wwwmcgrawhillcalinks
foundations11 and ollow the links
Key Concepts
bull Geometric shapes are used in architecture art and fashion These shapeoften use the golden ratio which is approximately 16181 The golden
ratio is used in design because its proportions are pleasing to the eye
bull Some geometric shapes can be used to ldquotile a planerdquo such that there are
no spaces These tiling patterns are known as tessellations Some polygon
that tile a plane are squares rectangles equilateral triangles and regular
hexagons Combinations of regular polygons can also tile a plane
Discuss the Concepts
D1 How are geometric shapes used in the construction of your school
Make a list of the shapes that you notice and where they are usedD2 Can the shape shown be used to tile the plane Use a sketch to expla
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
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For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1860
Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1960
Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2060
c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1160
Practise A 1 Which o these regular polygons cannot be used to tile a plane
A triangle B square
C hexagon D octagon
2 Sandor claims that the screen on his new HD television is a golden
rectangle It measures 32 in by 18 in Is he correct Justiy your answer
3 Look at the buildings in your neighbourhood Identiy geometric
shapes that you see in the construction o these buildings In
particular describe the use o shapes other than squares or rectangles
4 Tink about musical instruments that you have seen Geometric shapes
are present in the design o many musical instruments List several
instruments and identiy the geometric 1047297gures used in their design
5 Te old railroad station in Petrolia ON is now the public library
Identiy the different two-dimensional and three-dimensionalgeometric 1047297gures that were used in its design
Apply BLiteracy Connect 6 Uploading personal digital pictures to the Internet is a popular
activity I you do this and identiy people in your pictures thesofware that operates the posting site can ldquolearnrdquo to identiy aces
It uses measurements and ratios similar to those you ound or the
Mona Lisa in Investigate 1 Once a ace is in the database other users
can employ the sofware to identiy individuals in other pictures
Write a short paragraph discussing the advantages and disadvantages
o having such a sofware tool reely available on the Internet
12 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1260
For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1460
Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1560
Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1660
Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1260
For help with questions 7 and 8 refer to Investigate A
7 Sports such as soccer or lacrosse are ofen played on rectangular
1047297elds Investigate the dimensions o several o sports 1047297elds Find
the dimensions o an Olympic swimming pool Are any dimensions
close to a golden rectangle
Technology TipYou can search for
images directly using
most search engines
rather than searching
for web sites and then
looking for images
8 Use the Internet to 1047297nd a picture o each o these amous buildingsCopy each picture into Te Geometerrsquos Sketchpad reg and use the
sofware to identiy any golden rectangles that were used in the desi
a) Te Parthenon in Athens Greece Use a ront view
b) Te aj Mahal in Agra India Use a ront view taken down the
re1047298ecting pool
c) Te United Nations building in New York NY Use a view rom
the river
d) Te Buddhist emple in Niagara Falls ON Use a ront view
9 Te actor Drew Barrymore is considered as a classic example o aldquoround acerdquo while actor Fred Gwynne (who played a Frankenstein-
like character on a popular television show) is considered to be an
example o a ldquolong acerdquo Search or pictures o each actor Use penci
and paper or Te Geometerrsquos Sketchpad reg to determine how the acto
aces compare to the golden ratio
10 Te aces o some animals appear to have a greater width than heigh
One example is a house cat Use pencil and paper or Te Geometerrsquos
Sketchpad reg to determine the acial ratio o a typical cat
61 Investigate Geometric Shapes and Figures bull MHR
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Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1660
Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2060
c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2960
Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3160
Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3460
Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1360
Chapter Problem 11 Paul has been reading about the history o architecture Ancient
Egyptian builders ofen needed to lay out large right angles or their
constructions Tey did this using a ldquo12-knot roperdquo Obtain a piece o
string about 1 m long ie 12 knots in the string such that the knots
are 5 cm apart ie the ends o the string together such that you have
12 equal sections Arrange the string to orm a triangle with sides o
3 4 and 5 sections a) Measure the angle opposite the longest side
b) Write a short explanation o the mathematics behind this method
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
12 Where is the geometry in music Te 1047297rst ew bars o the classic
hard rock piece ldquoIn-A-Gadda-Da-Vidardquo by Iron Butter1047298y widely
acknowledged as the 1047297rst heavy metal band are shown
c
1
a) Notice the groupings o our notes Draw lines joining adjacent
notes in each group and then a single line returning to the 1047297rst
note Te 1047297rst triangle has been drawn or you Describe the kinds
o triangles that are ormed
b) riangles are ofen present in music composition Find two three-
note congruent triangles in this short excerpt rom the popular
holiday song ldquoO annenbaumrdquo
1
34
13 A stained glass panel contains several rectangles Which rectangles
are golden rectangles or close to being golden rectangles Explain
how you know
40 mm160 mm
68 mm
42 mm
64 mm
32 mm
32 mm200 mm
14 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1860
Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2060
c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
7212019 Geometry in Design
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1460
Achievement Check
14 Anwar claims that a hexagon does not need to be regular to tile
the plane He says that any hexagon that has opposite sides equal
will work
a) Draw a hexagon with opposite sides equal Be sure to use three
different lengths in your drawing
b) Determine whether your hexagon can be used to tile the plane I
it can show how it can be done I it cannot explain why not
Extend C 15 An architect is designing a window using the golden ratio
a) Te longer side measures 25 m o the nearest tenth o a metre
what should be the length o the shorter side
b) Draw a sketch o the window Cut the window in hal using themidpoints o the lengths Compare the ratios o sides o one o th
smaller rectangles What do you notice
16 Many countries use international paper sizes such as ISO A4 A she
o A4 paper measures 297 cm by 210 cm Tis size is sold in Canad
at offi ce supply stores Obtain a piece o A4 paper or make one rom
a larger piece o paper
a) Find the ratio o the length to the width and record it
b) Fold the paper in hal across a line joining the midpoints o the
lengths Measure the length and width o the olded paper Findthe ratio o the length to the width
c) Fold the paper again Find the ratio o the length to the width o
the olded piece What do you notice
d) Te ratio o length to width or the ISO standard is always radic__
2 1
Use a calculator to compare this ratio to the ratios that you ound
17 Use pencil and paper or sofware such as Te Geometerrsquos Sketchpad
to design abric with a tessellation print Te abric should appeal to
person with a particular career or hobby such as a musician a gole
a pilot a nurse etc Be sure to use basic geometric shapes to orm amore complex shape Be creative in your selection o colours
61 Investigate Geometric Shapes and Figures bull MHR
7212019 Geometry in Design
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Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1860
Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1960
Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2060
c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2160
Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1560
Perspective and OrthographicDrawings62
scale model
bull a model that is an
enlargement of a small
object or a reduction of
a large object
Habitat 67 is an apartment complex in Montreal It was designed by
Canadian architect Moshe Sadie or Expo rsquo67 as an experiment in
building apartments that allowed urban dwellers to enjoy both privacy
and outdoor space A scale model was used to represent what the
complex would look like when 1047297nished
isometric perspectivedrawing
bull visual representation
of three-dimensional
objects in two
dimensions
orthographicprojection
bull a set of drawings that
show up to six views of
an object
bull usually front side and
top views are given
orthographic drawing
bull a drawing that
uses orthographic
projection
Another means o representing three-dimensional objects is an isometric
perspective drawing which creates the illusion o three dimensions on a
two-dimensional plane A third representation is a series o orthographic
projections which illustrate ront side and top views o the object
How could you draw the isometric perspective and the orthographic
representations o Habitat 67 How can drawing sofware be used to
create these representations
o learn more about Habitat 67 go to wwwmcgrawhillcalinks foundations11 and ollow the links
16 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1660
Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1760
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1860
Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1660
Investigate 1 Perspective Drawings of Cubes and Rectangular Prisms
Use a single linking cube or a model o a cube urn the cube until an
edge is acing you Ten tip the cube downwards as shown Tis is the
perspective that you will draw
Tools isometric dot paper
linking cubes or other
physical models ofcubes
ruler
coloured pencils
Method 1 Use Pencil and Paper
1 a) Inspect the isometric dot paper Find the lines o dots that are
vertical Draw a short vertical line segment in blue
b) Find the lines o dots that slope down to the right at 30deg Draw ashort line segment sloping down to the right at 30deg in red
c) Find the lines o dots that slope down to the lef at 30deg Draw a
short line segment sloping down to the lef at 30deg in green
d) Use the dot pattern to draw the isometric perspective as shown
2 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
The word ldquoisometricrdquocomes from the
Greek words ldquoisordquo
meaning ldquothe samerdquo
and ldquometricrdquo meaning
ldquomeasurerdquo
Math Connect 3 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Draw an isometric perspect
drawing o the prism
62 Perspective and Orthographic Drawings bull MHR
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Choose Open rom the File menu
Navigate to the directory that stores Te Geometerrsquos Sketchpad reg
1047297les on your computer Tis will likely be called Sketchpad and
be located in the Program Files directory Choose Samples then
Sketches and then Geometry Select and open the 1047297le Dot Papergsp
Select the button Isometric vertical
Technology TipYou can change the
spacing of the dots
using the slider marked
ldquodot spacingrdquo
2 Inspect the isometric vertical dot pattern Find the lines o dots that are
vertical Find the lines o dots that slope down to the right at 30deg Find
the lines o dots that slope down to the lef at 30deg Use the dot pattern
to create the isometric perspective drawing o the cube as shown
3 Re1047298ect Compare the isometric perspective drawing with the view o
the cube On an isometric perspective drawing the lengths on the
drawing are the same as the lengths on the real cube Check that the
sides o the cube on the drawing are o equal length
4 Use linking cubes to construct a model o a rectangular prism
that measures 1 cube by 2 cubes by 3 cubes Draw an isometric
perspective drawing o the prism
18 MHR bull Chapter 6
7212019 Geometry in Design
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Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1860
Investigate 2 Orthographic Drawings of Cubes and Rectangular Prism
Tools square dot paper
linking cubes or other
physical models of
cubes
ruler
Method 1 Use Pencil and Paper
1 Use a single linking cube or a model o the cube urn
Front Vithe cube until the ront is acing you Draw the ront
o the cube Label the drawing
2 urn the cube until the right side is acing you Draw and label the
side view Similarly draw and label the top view
3 Re1047298ect What do you notice about the three orthographic drawings
o the cube A cube is a simple shape However the same method
can be used to represent more complex shapes
4 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
Tools computer
The Geometerrsquos
Sketchpad reg
Method 2 Use The Geometerrsquos Sketchpad reg
1 Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Select the button Square dot paper
2 Use a single linking cube or a model o a cube urn the cube until
the ront is acing you Draw the ront o the cube Label the drawin
3 urn the cube until the side is acing you Draw and label the side
view Similarly draw and label the top view Reer to the diagrams in
Method 1
4 Re1047298ect What do you notice about the three orthographic drawings o
the cube A cube is a simple shape However the same method can
used to represent more complex shapes
5 Use linking cubes to construct a model o a rectangular prism that
measures 1 cube by 2 cubes by 3 cubes Create three orthographic
drawings o the prism ront view side view and top view
You have been using Te Geometerrsquos Sketchpad reg to draw perspective
diagrams echnical designers use more sophisticated programs such as AutoCADreg Te science-1047297ction television series Babylon 5 was created
using AutoCADreg to render three-dimensional spaceships space station
and even some aliens Amateur 1047297lmmakers ofen use commonly availab
drawing and design sofware to create inexpensive special effects or
their productions
o learn more about drawing sofware go to wwwmcgrawhillcalinks
foundations11 and ollow the links
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
7212019 Geometry in Design
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 1960
Example 1 Modelling a Flight of Stairs
a) Use linking cubes to create a model or a 1047298ight o stairs Tere are
three steps Te steps are 30 cm deep and 120 cm wide Each riser
measures 30 cm Let the side o a cube represent 30 cm
b) Draw an isometric perspective drawing o the model
c) Sketch a set o orthographic drawings o the model
Solution
a) Te side o a cube represents 30 cm Te stairs will be 4 cubes wide
and 3 cubes deep at the bottom layer
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create the
isometric perspective drawing o the stairs
20 MHR bull Chapter 6
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c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2060
c) Use square dot paper or Te Geometerrsquos Sketchpad reg to draw the
orthographic drawings or the stairs
ront view
top view
side view
Example 2 Isometric Drawing of a Square-Based PyramidConstruct a model o a square-based pyramid using drinking straws
and masking tape or other materials Use isometric dot paper or Te
Geometerrsquos Sketchpad reg to sketch an isometric perspective drawing o
the pyramid
Solution
First draw the base Te base will appear as a diamond shape Select a
point in line with two opposite vertices to be the apex o the pyramid
Complete the pyramid by drawing the sides
62 Perspective and Orthographic Drawings bull MHR
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Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2160
Example 3 Isometric Drawings With Curved Edges
Consider a concrete model o a cylinder such as a juice or soup can
Use Te Geometerrsquos Sketchpad reg to sketch an isometric perspective
drawing o the can
SolutionCurved edges are represented by arcs o circles in isometric
perspective drawings
bull Open Te Geometerrsquos Sketchpad reg Open the 1047297le Dot Papergsp
Press the button Isometric vertical
bull Use the Point ool to draw three points just to the lef o three
o the points on the paper as shown
Technology Tip The dots in the
dot paper sketch
in The GeometerrsquosSketchpad reg are drawn
using transformations
If you try to draw an
arc through three of
these dots it will not
work Try it
bull Select the three points Select Arc Trough 3 Points rom the
Construct menu Drag the arc slightly to the right such that it alls
on three points on the dot paper In a similar manner construct
three more arcs Ten use the Segment tool to complete the sideso the cylinder as shown
Note You can also draw an isometric perspective drawing o a cylinder
on paper using a ruler and a compass
22 MHR bull Chapter 6
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Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2260
Key Concepts
bull A three-dimensional object can be represented in three ways
ndash as a scale model using concrete materials
ndash as an isometric perspective drawing
ndash as a set of orthographic drawings (usually the front view side view
and top view)
Discuss the Concepts
D1 Consider the isometric perspective drawing shown Indira claims
that all three orthographic drawings of this object will look the
same Laszlo says that only the front and the side will look the
same Soon-Tek says that all three will be different Who is correct
Give reasons
D2 Consider the isometric perspective drawing shown Discuss
whether the drawing could represent a real object Give reasons
for your answer
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2460
Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
7212019 Geometry in Design
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2360
Practise A 1 Which o these representations is not suitable or representing a
three-dimensional object Explain why
A a scale model
B an isometric perspective drawing
C a blueprint showing ront and side views
D a set o orthographic drawings
2 Marcie is building a scale model o an offi ce building using linking
cubes Te building is a rectangular prism with a length o 49 m a
width o 35 m and a height o 56 m Suggest a suitable scale and
calculate the dimensions or the model
3 Farmer Jacques is building a new silo in the shape o a cylinder o
maximize the storage volume he is planning to make the diameter
the same as the height Which o these diagrams matches the side
view
A B C D
4 Ivanka is using linking cubes to make a scale model o the isometric
perspective drawing shown Describe the physical model in terms olength width and height
24 MHR bull Chapter 6
7212019 Geometry in Design
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Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2760
Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2860
The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2960
Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2460
Apply BLiteracy Connect 5 Study the isometric perspective drawing shown Write a short
paragraph explaining why this drawing cannot represent a real obje
For help with question 6 refer to Example 1
6 Central courtyards are eatured in the architecture o many cultures
Te Romans ofen built their houses or villas with a central open
space surrounded by the rooms o the house a) Use linking cubes or other concrete materials to build a scale
model o a Roman villa Include a set o rooms surrounding a
central open space
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
isometric perspective drawing o the model
c) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
set o orthographic drawings or the model
For help with question 7 refer to Examples 2 and 3
7 Salvatore needs an isometric perspective drawing o a cone such tha
the height o the cone appears equal to the diameter
a) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create
an isometric perspective drawing o a cone that closely matches
Salvatorersquos requirements
b) Is it possible to match the height to the diameter exactly i
the diameter and the tip o the cone must land on one o the
dots Explain
62 Perspective and Orthographic Drawings bull MHR
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2960
Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3160
Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3360
9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3860
Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
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Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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8 Te artist M C Escher ofen used alse perspective in his drawings
to depict impossible situations Study the Escher work Convex
and Concave careully to see how the illusions work Consider the
ldquostairwayrdquo at the centre right o the drawing
a) Approximate the stairway with a triangle (ie do not show the
individual steps) Create a set o orthographic drawings in which
you see this as a stairway going up b) Create a set o orthographic drawings in which you see this as the
bottom support o an arch
c) Describe the triangles that occur in your orthographic drawings
Chapter Problem 9 Paul has been given a set
ront view top view side view
o orthographic drawings
or an inn to be built
near a tropical beach
Te design incorporates
open spaces throughoutthe building or air circulation Paulrsquos task is to create a scale model
and an isometric perspective drawing o the inn
a) Use the orthographic drawings to construct a scale model using
linking cubes or other concrete materials
b) Use isometric dot paper or Te Geometerrsquos Sketchpad reg to make an
isometric perspective drawing o the inn
Achievement Check
10 Sara used linking cubes to build the
structure shown in isometric perspective
Which drawing is a possible
orthographic projection o the top
view o the structure Explain
A B
C D
Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
26 MHR bull Chapter 6
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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Extend C 11 Use isometric dot paper or Te Geometerrsquos Sketchpad reg to create an
ldquoimpossiblerdquo isometric perspective drawing different rom the ones
in question D2 and question 5
12 In isometric perspective drawings the lengths in the drawing o a
cube match the side lengths o the real cube In real lie the apparen
size o an object decreases as the object moves arther away rom an
observer Artists use the concept o a ldquovanishing pointrdquo to achieve th
effect o perspective in their art
o see how this works draw a square using square dot paper or Te
Geometerrsquos Sketchpad reg Select a point ar away rom the square as
shown Join the vertices o the square to this ldquovanishing pointrdquo
Use these lines to draw the rest o the cube in perspective Ten hid
or erase the parts o the lines that extend outside o the cube
13 Advertisements or buildings that have not yet been constructed
ofen make use o two-point perspective In the example shown eac
street leads to a vanishing point Te lef side view o the building
uses the lef vanishing point while the right side view uses the right
vanishing point
Use square dot paper or Te Geometerrsquos Sketchpad reg to make a
two-point perspective drawing o the model in question 4
62 Perspective and Orthographic Drawings bull MHR
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Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2860
The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2760
Create Nets Plans and Patterns63
Commercial products are packaged in various shapes o containers Tese
containers are ofen manuactured as a two-dimensional net and then
olded or bent into a three-dimensional shape One example is a cereal
box which is a rectangular prism
Other three-dimensional objects such as houses or items o urniture are
constructed rom a plan Plans are usually parts o nets drawn to scale
that give enough inormation to construct the three-dimensional objectClothing is made by using a pattern to cut the cloth Te cloth pieces are
then sewn together to make the clothing item A pattern is usually a net
that is drawn in separate pieces
Investigate 1 Design a Box
Tools empty cereal box
scissors
ruler
tape
Many products are packaged in rectangular prisms
1 Measure and record the dimensions Cereal
l w
ho the cereal box
net
bull a two-dimensional
diagram that can be cut
out and folded to form a
three-dimensional object
plan
bull a scale drawing of astructure or object
bull a design or arrangement
scheme
pattern
bull a form template or
model from which an
object can be created
28 MHR bull Chapter 6
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 2960
Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
7212019 Geometry in Design
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3160
Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3360
9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3460
Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
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1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
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Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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The 1047297rst of the
modern packaged
cereals was toasted
corn 1047298akes introduced
in 1906 To learn more
about packaging
go to httpwww
mcgrawhillcalinks
foundations11 and
follow the links
Math Connect 2 Find the edge on the cereal box that was used to join the cardboard
Cut along this edge Cut any other joins that you 1047297nd
3 Unold the cereal box to orm a net Measure the edges o the net
Identiy each edge as one o the dimensions o the box
4 Re1047298ect Compare the measures o the net to the measures o the cere
box How are the dimensions o the net ound to produce a box o thdesired shape and size
5 Sketch a net or a box that will measure 4 cm by 3 cm by 2 cm Labe
the edges est your net by cutting it out and olding it into a box
ape the box together Measure the dimensions to check that the
box matches the dimensions given Suggest a product that would be
suitable or a package o this size
Investigate 2 Design a CylinderTools empty tissue roll
ruler
tape
Some products are packaged in cylindrical containers
1 Measure and record the height and diameter o the tissue rolld
h
2 Cut the roll along its height and spread it into a rectangle race
the rectangle onto a piece o paper Measure the dimensions o the
rectangle Which dimension is equal to the height o the roll What is
the name o the other dimension in relation to the cylinder Divide th
other dimension by the diameter o the tissue roll What do you notic
3 o draw two congruent circles or the top and bottom o the cylinde
what dimension do you need to consider Draw two circles or the
top and bottom using this dimension Cut out the net and bend it t
orm a cylinder with a top and a bottom
4 Re1047298ect Compare the measures o the net to the measures o the
cylinder How are the dimensions o the net related to the height an
diameter o the cylinder
5 Sketch a net that can be bent into a cylinder with a diameter o 6 cm
and a height o 10 cm est your net by cutting it out and bending it in
a cylinder with a top and a bottom ape the model together
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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Investigate 3 Patterns
Tools simple item of clothing
(old T-shirt etc)
scissors
newsprint
Patterns or clothing are nets that are used to cut cloth Te pieces o cloth
are sewn to orm the three-dimensional garment
1 Identiy the seams on the article o clothing
Use scissors to careully cut the threads along
each seam Lay out the pieces
2 Sketch the pattern on newsprint
3 Re1047298ect How can clothing manuacturers use a pattern to cut out
hundreds o pieces at one time Research clothing manuacturing
on the Internet to help you explain
4 Sketch a pattern that could be used to
cut out the pieces or a bucket hat
Example Cabin Plan
a) Draw a 1047298oor plan or a vacation cabin Te cabin will measure 8 m
along the side by 6 m across the ront Tere will be two congruent
square bunkrooms at the back with side length 3 m Tere will be a
bathroom measuring 2 m by 3 m along the ront wall Te remainingspace will be living area Label the measurements on your plan
b) Te cabin will be o the A-rame type measuring 5 m tall
Sketch orthographic drawings o the cabin Label all measurements
30 MHR bull Chapter 6
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Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3060
Solution
a) Select a suitable scale For example let 1 cm represent 1 m Draw a
rectangle to represent the 1047298oor o the cabin Draw walls or the two
bunkrooms and the bathroom Label all dimensions
6 m
3 m
3 m8 m
5 m
2 m
3 m
3 m
3 m
Bunk RoomBathroom
Bunk Room
3 m
b) Draw the ront side and top views o the cabin
6 m 6 m 6 m
8 m
5 m5 m
ront view top view side view
Key Concepts
bull A three-dimensional object can be constructed by folding or bending
a two-dimensional net
bull Parts of a net drawn to scale can be used as a plan to construct a
three-dimensional object
bull A net can be separated into pieces to form a pattern The pattern
can be used to make pieces of a three-dimensional object which
can be assembled into the object
Discuss the ConceptsD1 Sketch a net for a cube Explain how it must be folded to make
the cube
D2 A hobby club builds and 1047298ies model airplanes To make such an
airplane would you use a net a plan or a pattern Discuss the
advantages and disadvantages of each representation
63 Create Nets Plans and Patterns bull MHR
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3360
9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3860
Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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Practise A 1 Consider the net shown Te distance
between two dots represents 5 cm When
olded what three-dimensional object
will the net make
A an open box 5 cm high with a square
base with side length 15 cm
B a closed box 5 cm high with a square
base with side length 15 cm
C an open box 15 cm high with a square
base with side length 5 cm
2 Draw a net that can be used to make a cube-shaped box with an
open top
3 Sketch a pattern that could be
used to make a mitten
4 Woodworking magazines ofen include diagrams in their articles
about building urniture I you were writing an article about building
a bookcase would you include a net a plan or a pattern Explain
Apply B
For help with question 5 refer to Investigate 1Literacy Connect 5 Neela submitted an alternative net or a cube as shown
a) Write a short paragraph explaining why Neelarsquos net will not work
b) Explain how the net can be modi1047297ed so it will work
32 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3260
For help with question 6 refer to Investigate 2
6 A bakery needs a new cylindrical package or their oatmeal cookies
Each cookie is 1 cm thick and has a diameter o 8 cm Each package
should hold a dozen cookies
a) Draw a net that can be used to make a model o the package
b) Cut out your net and bend it to orm the package ape the mod
together
c) Check the dimensions o the model by measuring Do they meet
the requirements
7 An imported cheese is sold in the
4 cm
4 cm
4
12 cm
shape o a triangular prism Te ends
o the prism are congruent equilateral
triangles Each triangle has a side
length o 4 cm and the prism is
12 cm long
a) Draw a net that can be used to make
a model o the package
b) Cut out your net and bend it to
orm the model ape the model together
c) Check the dimensions o the model by measuring Do they meet
the requirements
Chapter Problem 8 Paulrsquos company has been asked to design a grain storage shed that
will be exported to developing countries Te shed will be made o
galvanized sheet metal with our congruent square walls with side
length 3 m Te roo will be our congruent equilateral triangles wit
side length 3 m Te bottom will be lef open so that the shed can be
attached to a poured concrete base Te sheds will be shipped as net
to conserve shipping space Paulrsquos task is to design the net or the
shed and to estimate the cost o the metal required
3 m
3 m
3 m3 m
a) Draw a net that can be used to make a model o the shed
b) Find the total area o sheet metal required
c) Sheet metal costs $650m2 Find the cost o the metal or the she
63 Create Nets Plans and Patterns bull MHR
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9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3860
Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3360
9 Te owner o a garden centre needs a pedestal or displaying a prize-
winning 1047298owering plant Te pedestal is to be made o painted sheet
metal with dimensions as shown Te bottom o the pedestal will be
lef open Draw a net that could be used to make the pedestal Show
all measurements
20 cm
10 cm
10 cm
Achievement Check
10 Reer to question 9 Te pedestal will be modi1047297ed Now the top
will have a circular hole to accommodate a 10-cm diameter metal
cylinder Te height o the cylinder will be 30 cm Draw the net
or the modi1047297ed pedestal and the net or the cylinder Show
all measurements
Extend C 11 A regular tetrahedron is a three-dimensional shape with our
congruent equilateral triangular aces Consider a tetrahedron
made o equilateral triangles with side length 3 cm
a) Draw a net that can be used to make the tetrahedron
b) Cut out your net and old it into a tetrahedron ape thetetrahedron together
12 Lydia made a square shape using eight linking
cubes leaving the centre space empty Draw a
net that could be used to make a paper model
o the shape I you think it is not possible to
make a net explain why
13 A regular octahedron has eight congruent
equilateral triangular aces It looks like twosquare-based pyramids glued together at their
bases Draw a net that can be used to make a regular
octahedron Cut out your net and old it into an
octahedron ape the octohedron together
34 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3460
Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
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1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
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Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Draw Plans WithDynamic Measurements
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a scale 1047298oor plan or the vacatiocabin in Example 1 o section 63 Te cabin will measure 8 m along
the side by 6 m across the ront Tere will be two congruent square
bunk rooms at the back with side length 3 m Tere will be a bathroom
measuring 2 m by 3 m along the ront wall Te remaining space will be
living area
1 Start by constructing the 1047297xed boundaries o the cabin Open Te
Geometerrsquos Sketchpad reg Choose Preferences rom the Edit menu a
then the Units tab Choose Show Grid rom the Graph menu Drag
the origin to the lower lef area o the workspace I necessary adjus
the unit point until you can see at least 10 gridlines horizontally and8 gridlines vertically
2 Choose Plot Points rom the Graph menu Plot the point (8 6) In the
same way plot the points (0 6) and (8 0) Select the origin and the thr
plotted points moving clockwise Choose Segments rom the Constr
menu You now have a rectangle that represents the boundaries o the
cabin ry to move one o the corners What happens
3 Use the Segment ool to sketch the bunk rooms and the bathroom
Select the endpoints o the width o a bunk room Choose Coordin
Distance rom the Graph menu Move the measurement to a
convenient location Measure and label dimensions or remaining th
rooms Include the outside dimensions o the cabin Hide the axes a
the grid
Use Technology
Use Technology bull MHR
7212019 Geometry in Design
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4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
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Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
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Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3560
4 Re1047298ect Select the line segment that orms the wall between
the bunk rooms Move it up and down Note how the measurements
change as you move the line segment ry changing the dimensions othe other rooms
5 Calculate the area o each room Choose Calculate rom the Measure
menu A calculator will appear Select the length o one bunk room
Press the button Select the width o the bunk room Move the area
measurement to a convenient location Change the label to Bunk
Room 1 Measure the areas o the other rooms o 1047297nd the living
area calculate the area o the entire cabin and subtract the areas o
the rooms
6 Re1047298ect Change the dimensions o one room by dragging a wallNotice how the measurements including the areas change
36 MHR bull Chapter 6
7212019 Geometry in Design
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Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3660
Scale Models64
A scale model o a building a piece o urniture or an item o clothing
is useul in the planning stages o construction or manuacturing Te
model can highlight problems that are not obvious in a two-dimensiona
representation Te model can also demonstrate improvements to a plan
Fixing the plan beore production can save materials resources and tim
Example 1 Design a Cylindrical Tank
An engineer is designing a cylindrical tank or a tanker truck that will
hold about 800 f3 o liquid Te length o the cylinder should be twice
the diameter and should be a whole number o eet
a) Use the ormula or the volume o a cylinder to determine the
dimensions o the tank required
b) Select a suitable scale and draw a net or a paper model o the tank
c) Cut out your net and bend it into the cylindrical shape ape the
model together
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3760
Solution
a) Te ormula or the volume o a cylinder is V = πr 2h In this case h is
twice the diameter or our times the radius
Te volume ormula becomes
V = πr 2(4r )
= 4
πr
3
ry different values or the radius
Radius (f)
1
2
3
4
5
Volume (f3)
126
1005
3393
8042
15708
A radius o 4 f gives a volume o just over 800 f3 Using a radius o
4 f the diameter is 2 times 4 f or 8 f and the length is 2 times 8 f or 16 f
b) One dimension o the rectangle or the net is the length o the
cylinder or 16 f Te other dimension is the circumerence o
the cylinder
C = πd
= π(8)
251
Te circumerence is approximately 251 f
Select a suitable scale (or example
1 in represents 4 f) Draw therectangle or the with a net Add
two circles each diameter o
2 in or the ends
c)
38 MHR bull Chapter 6
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
7212019 Geometry in Design
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
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1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
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Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
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Example 2 Design a Pentagonal House
An architect has designed a house in the shape o a regular pentagonal
prism Te 1047298oor orms a pentagon with a side length o 6 m Te 1047297ve
walls are congruent squares Te peaked roo is made o 1047297ve congruent
equilateral triangles Select a suitable scale Draw a net that can be used
make a scale model o this house Ten cut out your net and old it intomodel o the house ape the model together
Solution
Method 1 Use Pencil and Paper
bull Choose an appropriate scale For example 1 cm represents 1 m Draw
a regular pentagon A regular pentagon consists o 1047297ve congruent
isosceles triangles whose apexes meet at the centre o the pentagon
Te apex angle or each triangle is 360deg divide 5 = 72deg Draw a dot or the
centre and a horizontal line segment to the 1047297rst vertex Draw an angle
o 72deg rom the horizontal line segment Use compasses to help you
create an isosceles triangle that includes the second vertex Continue
this process until you have 5 vertices Join the vertices
bull Use a ruler and compasses to draw a square on each side o the
pentagon Ten draw an equilateral triangle on each square
bull Use your net to make a
scale model o the house
Method 2 Use The Geometerrsquos Sketchpad reg
bull Draw the pentagonal base urn on automatic labelling o points Sket
a horizontal line segment AB Select point A and choose Mark Cente
rom the ransform menu Select the segment and point B ChooseRotate rom the ransform menu Change the 1047297xed angle to 72deg
Select Rotate Continue until you have 1047297ve vertices Join the vertices
bull Draw the square walls Select one endpoint o a side Choose Mark
Center rom the ransform menu Select the segment and the other
endpoint Choose Rotate rom the ransform menu Change the 1047297xe
angle to 90deg Select Rotate Repeat or each side Ten use the other
64 Scale Models bull MHR
7212019 Geometry in Design
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s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
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a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
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a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
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Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
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1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
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Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 3960
s
t
1 m
1 m
1 m
3 m
endpoint o the side as the centre and rotate through an angle o 270deg
Complete the 1047297ve squares
bull Draw the triangular roo aces Select one outside corner o a square
Choose Mark Center rom the ransform menu Select the outside
segment and the other endpoint Choose Rotate rom the ransform
menu Change the 1047297xed angle to 60deg Select Rotate Repeat or the other
squares Complete the 1047297ve triangles Hide the points and labelsbull Print your net and use it to make a scale model o the house
Example 3 Design a Tent
An out1047297tting company plans to manuacture a new tent Orthographic
drawings or the tent are shown Te distance between pairs o horizontal
or vertical dots represents 1 m
s
t
ront view top view side view
Use paper or drinking straws and masking tape to construct a scale model
o the tent Select a suitable scale Make any necessary calculations
SolutionLet s represent the slant length o the side o the tent and t represent the
slant length o the roo Use the Pythagorean theorem to 1047297nd the lengths
o s and t
s2 = 32 + 12 t 2 = 12 + 12
s2 = 10 t 2 = 2
s = radic ___
10 t = radic __
2
s 32 t 14
Te side slant length is Te top slant length is
approximately 32 m approximately 14 m
Select a scale that is suitable or your materials For example let 4 cm
represent 1 m Te base o the model will measure 4 times 4 cm or 16 cm
Te side slant length will measure 4 times 32 cm or 128 cm Te top slant
length will measure 4 times 14 cm or 56 cm
40 MHR bull Chapter 6
7212019 Geometry in Design
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Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
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1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
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Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4060
Key Concepts
bull Scale models can be constructed using nets orthographic drawings
or isometric perspective drawings
bull Models are useful in the planning stages of a project
bull Models can identify problems in a plan or suggest improvements
Discuss the Concepts
D1 Can a model of any three-dimensional object be constructed
using a net Explain
D2 To obtain a building permit for a house the owner must submit a
set of orthographic drawings known as elevations Suggest reasons
why orthographic drawing are required rather than a scale model
or an isometric perspective drawing
Practise A 1 Maria is building a scale model o a garden shed She will let 1 in
represent 2 f I the base o the shed measures 10 f by 12 f what
measurements will Maria need or the model
2 Te most amous Canadian-designed and -built aircraf was the Av
CF-105 Arrow A number o 1 _
8 scale models were made or testing i
a wind tunnel Te length o the model was 10 f 8 in What was the
length o the ull-size aircraf
3 Te town o Port Colborne is planning to construct a new
recreational centre including a skating arena and a swimming pool
Suggest reasons why an architect bidding on the contract wouldprovide a scale model in addition to drawings
4 Freda is planning a new silo or her arm made o a cylinder topped
with a hemisphere Te height o the cylinder will orm a golden rat
with its diameter Freda would like to make a scale model o the silo
to see i her design works Describe a procedure and suggest materi
that Freda could use to make the scale model
64 Scale Models bull MHR
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5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
7212019 Geometry in Design
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4160
5 Suleiman plans to construct a
ldquominimalistrdquo chair as shown in
the isometric perspective drawing
He will use three pieces o
1 in-thick plywood Te distance
between a pair o dots represents
1 f Draw plans that show the pieces owood required with measurements
Apply BLiteracy Connect 6 A standard gol ball has a diameter o 168 in You have been
contracted to design a package to hold 12 gol balls Sketch a set
o orthographic drawings or your chosen package Write a brieexplanation o why you selected this design
7 Draw an isometric perspective drawing or the tent in Example 3
8 A local artist wants to include a rectangular prism in her newest
sculpture Te prism should have a length o 1 m a width o 2 m
and a height o 3 m
a) Select a suitable scale Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale model
o the prism
b) Cut out your net and old it to make the model ape the model
together
c) Te artist has decided to paint all o the sides o the prism except
one which will be placed on the ground She has not however
decided which side will be placed on the ground I paint costs
$2m2 what are the possible costs or painting the prism
For help with question 8 refer to Example 1
9 A new soup can must hold at least 500 mL o soup Its height must
be equal to its diameter Te height and the diameter must be wholenumbers o centimetres
a) Determine the minimum height and width o the can
b) Select a suitable scale Draw a net that can be used to make a scale
model o the can
c) Cut out your net and bend it to make a scale model o the can
ape the model together
42 MHR bull Chapter 6
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For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
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Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4260
For help with question 9 refer to Example 2Reasoning and Proving
Re1047298ecting
Selecting Tools
Connecting
Representing
Problem Solving
Communicating
10 A band council has decided to commission a new community centr
to be used or various events such as the annual pow-wow Te cent
will be in the shape o a regular hexagon with each side measuring
10 m Each congruent rectangular wall will be 3 m high Te buildin
will have a 1047298at roo
a) Select a suitable scale Choose measurements that ensure youhave a regular hexagon Use pencil and paper or Te Geometerrsquos
Sketchpad reg to draw a net that can be used to make a scale mode
o the centre
b) Cut out your net and old it to make the model ape the model
together
c) Te base o the centre will be poured concrete 10 cm thick ake
measurements rom your net to 1047297nd the area o the hexagonal
base Multiply the area by the thickness to determine how many
cubic metres o concrete will be needed d) Concrete costs $75m3 delivered to the site Estimate the cost o
the concrete or the base
11 Afer studying the scale model in question 10 the band council
decided to change the 1047298at roo to one made o six congruent isoscel
triangles Tis will ensure that snow and rain will not accumulate on
the roo It will also make the building more attractive Te central
peak o the roo will be 5 m above the 1047298oor
a) How will this change affect your net Explain
b) How will this change affect the cost o concrete or the base Explai
Chapter Problem 12 A customer has seen the post offi ce in Gorrie ON and would like
a similar building or his business He has supplied Paulrsquos company
with sample orthographic drawings as shown Te distance betwee
pairs o horizontal or vertical dots represents 2 m
a) Draw a net that can be used to make a scale model o the buildin
b) Cut out the net and construct the model ape the model togeth
c) Te cost o the building is estimated to be $1200m2 o 1047298oor area
Estimate the cost o the building
64 Scale Models bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
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4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
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7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
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9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4360
Achievement Check
13 A propane storage tank is to be built rom the orthographic drawings
shown Te distance between pairs o horizontal or vertical dots
represents 1 m
Which description best 1047297ts the propane tank Explain
A A cylinder 11 m long with a hemisphere o radius 2 m at
each end
B A cylinder 7 m long with a hemisphere o radius 2 m at each end C A cylinder 11 m long with a hemisphere o radius 4 m at
each end
D A cylinder 7 m long with a hemisphere o radius 4 m at each end
Extend C 14 How can you make a net or a cone Roll a piece o notebook paper
to orm a horn Cut the horn to orm a cone Continue cutting until
you have the minimum amount o paper necessary to orm the cone
Unroll the paper and sketch the net
15 Use trial and error to make a net or a cone with a radius o 5 cm
and a height o 3 cm Cut out your net and use it to make the cone
Measure the dimensions to check that they are correct
44 MHR bull Chapter 6
7212019 Geometry in Design
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Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4460
Solve Problems With GivenConstraints65
constraint
bull a condition that limits
the acceptable range
of values for a variable
Designers such as architects engineers or ashion designers ofen must
ensure their designs satisy given constraints Examples o constraints
include a maximum cost a minimum or maximum size or regulations
governing saety energy effi ciency or perormance
Example 1 Modular House
Orthographic drawings or a modular house are shown
s
tu
ront view top view side view
Te distance between pairs o vertical or horizontal dots represents
5 f Te house will be built in two sections and then transported to the
building site or assembly Highway traffi c rules limit the width o each
section to 15 f or less Te centre wall will be located 15 f rom the lef
ront wall It will be doubly thick so that each hal o the house will hav
our walls during transport
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4560
a) Select a suitable scale Draw a net or each section o the house
b) Cut out your nets and make scale models or the two sections o the
house ape your models together Ensure that the models 1047297t together
properly or assembly
Solution
a) From the orthographic drawings the centre wall is 175 f high Usethe Pythagorean theorem to determine the slant lengths o the roo
s
t u
15 f 5 f 5 f
25 f
75 f5 f
s2 = 152 + 752 t 2 = 52 + 252 u2 = 52 + 52
s2 = 28125 t 2 = 3125 u2 = 50
s 168 t 56 u 71 Slant length s is Slant length t is Slant length u is
approximately 168 f approximately 56 f approximately 71 f
Let the distance between pair o horizontal or vertical dots represent
5 f Calculate the measures or the scale drawing by dividing each
actual measure by 5
Te net or the lef side o the house
168 f 10 f
10 f
10 f
175 f
175 f
175 f
40 f
15 f
46 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
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Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
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Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4660
Te net or the right side o the house
175 f
175 f
175 f
40 f
10 f 15 f
15 f
15 f
71 f
71 f
71 f
56 f
56 f
56 f
b) Use the nets to make scale models o the two halves o the house
o learn more about modular homes go to wwwmcgrawhillcalink
foundations11 and ollow the links
Example 2 Oil Tank
Sometimes basic saety considerations depend on understanding nets
and volumes Consider an engineer who needs to design a square-based
berm a shallow container to prevent the spread o oil rom a leaking oil
tank Te cylindrical oil tank is 20 m in diameter and has a height o
20 m Te berm must have a height o 5 m
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
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9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4760
a) Make appropriate calculations and 1047297nd the minimum dimensions
o the berm
b) Select a suitable scale Draw a net or the berm and a net or the oil
tank Use your nets to construct a scale model o the arrangement
required
Solution a) Start by calculating the maximum volume o oil that must be
contained i the tank is ull
V = πr 2h
= π times 102 times 20
6283
A square-based
berm has the
shortest side
lengths requiredto contain a
given volume
Minimizing the
bermrsquos perimeter
cuts down on the
cost of materials
Math Connect Te berm must be able to hold 6283 m3 o oil
Let the side length o the square base o the berm be represented by s
Volume o berm = area o base times height
= s2 times 5
= 5s2
5s2 = 6283
s2 = 6283
_
5
s2 = 12566
s = radic ______
12566
s 354
It is saer to overestimate Round 354 m up to 40 m Suitable
dimensions or the berm would be 40 m by 40 m by 5 m b) Draw a net or the berm Let the space between pairs o horizontal or
vertical dots represents 5 m Te base o the berm measures 8 dots by
8 dots and the height measures 1 dot
48 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4860
Draw a net or the oil tank Te net consists o two circles o
diameter 20 m or the top and bottom plus a rectangle or the wall
o the cylinder One dimension o the rectangle is the same as the
height o the tank or 20 m Te other dimension is the same as the
circumerence o the tank
C = πd
= π times 20
63
Te circumerence is approximately 63 m
Use the nets to construct the scale models or the berm and the
oil tank
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 4960
Key Concepts
bull Design problems are often subject to constraints or conditions
bull Constraints may include cost considerations safety regulations
maximum or minimum sizes or performance requirements
Discuss the Concepts
D1 A fashion designer is to design a new line of sunglasses to
complement a clothing line Discuss some of the constraints
that she might face during the design stage
D2 The highest mountain in Africa is
Mt Kilimanjaro in Tanzania On the
Marangu hiking route to the top
climbers spend the night in huts
designed by Norwegian architects
There is no road The trail is the
only route to the huts Discuss some of the constraints that the architects faced when
designing the huts
Practise A 1 A new laundry detergent will be sold in a cubical box Te box holds
1 L o detergent Find the side length o the box
2 A actory temporarily stores liquid wastes in a circular cement-lined
pond that has a base area o 500 m2 Find the length o berm needed
to enclose the pond
3 Santorsquos Pizzeria is introducing a new giant-size pizza with a diameter
o 24 in Give possible reasons why the pizza will be packaged in an
octagonal box rather than a square box
4 Preabricated housing or developing countries with warm climates is
ofen made o galvanized sheet metal Discuss some o the constraints
that might affect the designers o this type o housing
50 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5060
Apply B 5 A one-story bungalow is to be designed or a small lot Te 1047298oor pla
o the house must 1047297t into a rectangle measuring 30 f by 40 f Te
buyer has made the ollowing speci1047297cations
Room Minimum Floor Area (ft2)
bedroom 1 200
bedroom 2 150
bedroom 3 100
bathroom 100
eat-in kitchen 200
living room 300
Tere must be a clear path to all rooms Use pencil and paper or
drawing sofware to design such a house Design ip Lay out the
rooms such that you minimize the area required or hallways
6 A 1047298exible 1047298oating pipe is used to contain an oil spill Te pipe can
contain a depth o 50 cm o oil It is placed around a leaking tanker
carrying 200 000 m3 o oil Te pipe is laid such that it orms a circle
around the leaking tanker What length o pipe is needed to contain
all o the oil
7 Build a boat using one piece o water-resistant cardboard or plastic
the same size as notebook paper up to six popsicle sticks glue as
required and enough tape to join the edges Te boat must hold the
maximum load possible without sinking est the boat using standa
weights coins or other materials Record your plan
Many bridges use
triangles in their
construction Triangles
are rigid and add to
the strength of the
bridge
Math Connect 8 Build a bridge using 12 plastic drinking straws one sheet o notebo
paper and duct tape to join the straws Te bridge must span 15 cm
and hold the maximum load possible Note Te duct tape may be
used only to join the straws It may not be used to reinorce the bridg
Place the bridge over a gap o 15 cm and test it with standard weights
coins or other materials Record your plan
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5160
9 Farmer Macdonald sketched an isometric perspective drawing o
a new chicken coop that he would like to build Te space between
pairs o dots represents 1 m
a) Draw a set o orthographic drawings or the coop b) Draw a net that can be used to make a scale model o the coop
Cut out the net and construct the model ape the model
together
c) All sides o the coop except the 1047298oor will be made o sheet metal
which sells or $15m2 Estimate the cost o buying the sheet metal
or the coop
Chapter Problem 10 Te uranium uel or a nuclear reactor is placed in a cylinder Te
cylinder lies horizontally on its side with a diameter o 8 m and a
length o 6 m It must be surrounded by a containment building in theshape o a vertical cylinder Paul has been given the task o designing
the building and estimating the cost o the buildings concrete exterior
a) Use pencil and paper or drawing sofware to determine the
minimum dimensions o the containment building required
b) Find the area o the walls and roo o the building Te concrete
exterior must be 80 cm thick Multiply the area by the thickness
to determine the approximate volume o concrete required in
cubic metres
c) Concrete can be delivered to the site and poured or $90m3
Estimate the cost o the concrete or the building
52 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5260
Achievement Check
11 Reer to the oil tank problem rom Example 2 Hartmut suggests
using a circular berm that is also 5 m high He says that the circular
berm would be smaller than the square berm Is he correct Show
calculations to support your answer
Extend C 12 Te inside o a new high-pressure scuba tank has a volume o 1 f3
Te walls top and bottom must be 05 in thick to withstand the
pressure Te ratio o height to diameter will be 31 Model the tank
using a cylinder Determine the dimensions required inside and
outside Prepare a set o orthographic drawings or the tank
13 An electric urnace can produce a maximum o 12 000 W o heat A
concrete building requires a heat source o 10 Wm2 on its exterior
walls and roo
a) Design a small offi ce building in the shape o a rectangular prism
that could be heated by this urnace running at maximum Note
For multiple storeys the building code requires that each storey
a minimum o 3 m in height
b) Construct a scale model o your building
14 Te layout or a new terminal at the international airport will use
the ldquosnow1047298akerdquo design Passengers will arrive through one ace
o a square-based reception building Once they have checked in
they will ollow a passageway through one o the other aces to ahexagonal ldquosnow1047298akerdquo gate cluster Five o the aces will be gates or
aircraf while the sixth will house the passageway Te aircraf usin
these gates need a clear square area with a side length o at least 50
in ront o the gate
Use pencil and paper or drawing sofware to design the terminal Te
passageways should be kept as short as possible to minimize the distan
that passengers need to walk to get rom one ldquosnow1047298akerdquo to another
Entrance
Check-in
Area
To Gates
Snow1047298akeGateCluster
To Gates
65 Solve Problems With Given Constraints bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5360
Use The Geometerrsquos Sketchpadreg to Create TessellationsUsing Rotations
Tools computer
The Geometerrsquos
Sketchpad reg
Use Te Geometerrsquos Sketchpad reg to draw a complex tessellation using
rotations Begin with a regular hexagon Ten alter the interior o the
hexagon using the ransform menu
1 Open Te Geometerrsquos Sketchpad reg Use the method shown in Section
64 Example 2 Method 2 to draw a regular hexagon All the angles
in each triangle will be 60deg Draw a point approximately in the centre
o one o the triangles that make up the hexagon Select this point as
well as the centre and a vertex o the hexagon as shown Choose Arc
Trough 3 Points rom the Construct menu
2 Select the centre o the hexagon Choose Mark Center rom the
ransform menu Select the arc Choose Rotate rom the ransform
menu Adjust the angle o rotation to 60deg Repeat the rotation until
you have six arcs one in each triangle
3 Select the six vertices o the hexagon Choose Hexagon Interior rom
the Construct menu You can change the colour by choosing Color
rom the Display menu Select one o the arcs Choose Arc Interior
rom the Construct menu Ten choose Arc Segment While the arc
interior is selected change the colour by choosing Color rom the
Display menu Repeat this procedure until you have six arc segments
o different colours
Use Technology
54 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5460
4 Hide the lines the points and the arcs
5 Select the hexagonal pattern and then choose Copy rom the Edit
menu Choose Paste rom the Edit menu Move the copy beside the
original pattern Continue pasting copies until you have ormed a
tessellation
6 Re1047298ect Tink o other ways to use rotations For example you can
use a polygon in place o an arc Consider other colour variations
What other regular polygons can be used or tessellations What
combinations o polygons can be used
Use Technology bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5560
61 Investigate Geometric Shapes
and Figures pages XXndashXX 1 Afer using the ldquo12-knot roperdquo to lay
out a square or rectangle on the ground
ancient builders then measured the two
diagonals Why did they do this Why
does this method work
2 Basiruddin has sewn a blanket or his
sisterrsquos new baby It measures 130 cm by
80 cm Basiruddin claims that the blanket
is a golden rectangle and that i it isolded in hal the hal-rectangle is also
a golden rectangle Use calculations to
check his claims
62 Perspective and OrthographicDrawings pages XXndashXX
3 Shawna is designing
an unusual seven-room
house or a client Each
room will have the
shape o a cube Shawna
has represented the
basic plan as an
isometric perspective
drawing as shown
a) Use linking cubes or other materials to
build a scale model o the house
b) Draw a set o orthographic drawings
or the house
4 Jorge drew orthographic drawings o
his amily home in Spain as shown Tedistance between horizontal or vertical
pairs o dots represents 2 m Draw
an isometric perspective drawing o
the house
ront view top view side view
63 Create Nets Plans and Patternspages XXndashXX
5 You are designing a rondaval a circular
hut Te hut will have a diameter o 24 f
In the centre will be a square washroom
block with side length 8 f Te hutrsquos
diameter will divide the washroom block
and the hut into two units each witha washroom and a bedroom Select a
suitable scale and draw a 1047298oor plan or
the rondaval Include all measurements
6 A new line o bath crystals will be
marketed in a decorative cylindrical
container with a diameter o 6 cm and a
height o 18 cm A cork will be used to
seal the container so no top is required
Review 6
56 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5660
Draw a net that could be used to make a
model o the container Cut out your netand construct the container Check the
dimensions o the model by measuring
7 David is building an airplane Te rame
is made o welded steel tubing and will
be covered with abric Each wing is in
the shape o a rectangle measuring 5 m
long by 1 m wide with a semicircle at
one end Te wing will be covered with
abric Sketch a pattern that could beused to sew an envelope to slip over the
wing rame Show all measurements
64 Scale Models pages XXndashXX
8 Model airplane kits are commonly sold
in1 _
72 scale A Boeing 777 airliner has a
wingspan o 609 m and a length o
739 m Find the wingspan and length o
a 1 _ 72
model o the Boeing 777
9 Quonset huts are ofen used or storage
o arm machinery aircraf or other
large machines
5 m
10 m
10 m
Sandrine is planning to build a Quonse
hut to shelter the ultralight aircraf thatshe is building It will have a square 1047298oo
with side length 10 m Te curved roo
will have a maximum height o 5 m
Draw a net that can be used to make a
scale model o the Quonset hut Cut ou
your net and construct the scale model
10 A new offi ce building will have a square
base with side length 40 m It will be
60 m high A revolving restaurant in thshape o a cylinder with a diameter o
80 m and a height o 4 m will be built
on top Draw nets that can be used to
make scale models o the building and
the restaurant Cut out your nets and
construct the models
65 Solve Problems With GivenConstraints pages XXndashXX
11 Martin is designing a hexagonalbunkhouse containing six triangular
rooms that meet at the centre Te base
o the bunkhouse will be poured concre
10 cm thick Concrete can be delivered
to the site or $75m3 Te budget or th
concrete base is a maximum o $600 T
walls will be congruent squares and the
roo will be 1047298at Te cost o wood or th
walls and the roo is $20m2
a) Design the base such that the
hexagon is as large as possible witho
exceeding the cost constraint
b) Continue the design or the walls
and roo Estimate the total cost o
materials or the base walls
and roo
Chapter 6 Review bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5760
1 Which rectangle A
D
E
G
B
C
F His closest tobeing a golden
rectangle
A ABCD B AEGD
C EBHF D FHCG
2 Te measures o the interior angles o
several regular polygons are shown
Which o these polygons can be used to
tile the plane A 108deg pentagon B 120deg hexagon
C 135deg octagon D all o them
3 An ornament is a square-based pyramid
with a sphere on top Te pyramid has a
base o side length 4 cm and a height o
6 cm Te diameter o the sphere is 5 cm
Which o these represents the top view o
the ornament
A B
C D
4 A new specialty ruit and nut bar is
being made Te bar has a diamondcross-section made up o two congruent
equilateral triangles with side length
12 cm Te bar is 24 cm long Draw a
set o orthographic drawings or the bar
Include all measurements
12 cm
12 cm
12 cm
24 cm
12 cm
12 cm
5 Select a suitable scale and draw a net
that could be used to make a scale model
o a package or the ruit and nut bar in
question 4
6 Yasmini received a pet budgie in acylindrical cage with a diameter o 60 cm
and a height o 50 cm Yasmini needs
a night cover or the cage Te cover
requires a top and sides but no bottom
Select a suitable scale and draw a pattern
that could be used to cut the pieces o
cloth needed to make the cover
Practice Test6
58 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5860
7 An isometric perspective drawing or
a new building is shown Te distance
between pairs o dots represents 5 m
Draw orthographic drawings or thebuilding
8 A gif box or a travel mug will have a
square base with side length 4 in and a
square lid with side length 6 in Te box
must be 5 in high Select a scale andsketch nets that can be used to make a
model o the box and the lid Show all
measurements on the net
9 A new tomato juice can is being designe
such that the ratio o the height to the
diameter is equal to the golden ratio T
diameter will be 12 cm
a) Select a scale and draw a net that
could be used to make a scale model
o the can
b) Determine the volume o tomato juic
that the can will hold
Chapter Problem Wrap-Up
Paul has learned a lot about geometry and
architecture during his coop placement Choose
a design project of your own similar to the projects
you have worked on throughout this chapter It
could be a building or structure a package or an
item of clothing
a) Write a plan for your project Include at least four
geometric elements in the design Describe the
constraints on your project and how you can meet them
b) Draw an isometric perspective drawing or a set of orthographic drawings
for your project
c) Calculate the projectrsquos dimensions Draw a pattern or a net Build a scale
model of your projectd) Write a summary of your project Include your drawings and model Show all
your calculations and measurements
Chapter 6 Practice Test bull MHR
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 5960
Chapter 4 Quadratic Relations I
1 a) Which o these relations are quadraticExplain your reasoning
i) x y
0 17
1 13
2 9
3 5
4 1
5 -3
6 -7
ii) x y
1 0
3 1
5 3
7 6
9 10
11 15
13 21
iii) y = 2x + 1 iv) y = 12 ndash 2x 2
b) Check your predictions by graphing
each relation
2 Describe the graph o each parabola
relative to the graph o y = x 2
a) y = (x + 4)2 b) y = ndash(x ndash 1)2
c) y = ndash03(x + 7)2 d) y = 2(x + 9)2
e) y = 09(x + 32)2 f) y = ndash4(x ndash 18)2
3 For each relation
i) identiy the coordinates o the vertex
ii) determine i the parabola opens
upward or downward
iii) determine i the parabola is vertically
stretched or vertically compressed
iv) sketch the graph
a) y = (x ndash 1)2 + 9 b) y = ndash2(x + 8)2 ndash 5
c) y = 01x 2 ndash 1 d) y = 91x 2
e) y = 05(x ndash 2)2 + 2 f) y = 8(x + 1)2 + 13
4 A 1047298are is sent up as a distress signal
Te path is modelled by the relationh = ndash49(t ndash 6)2 + 1774 where h is the
1047298arersquos distance above the water in metres
and t is the time in seconds
a) How long will the 1047298are take to reach
its maximum height What is the
maximum height
b) A typical 1047298are will stay lit or 7 s How
high above the water will the 1047298are be
7 s afer it is launched c) Graph this relation
d) Afer how many seconds will the 1047298are
hit the water
Chapter 5 Quadratic Relations II
5 Expand and simpliy
a) (3x + 4)(10x + 1) b) (x ndash 2)(4x + 15)
c) (12x ndash 8)(2x + 05) d) (6 + 2x )(6 ndash 2x )
6 a) Write an expression in simpli1047297ed
orm or the area o an apartment
with dimensions s ndash 1 by 4s ndash 7
b) I s = 15 m 1047297nd the actual area o the
apartment
7 Write each relation in standard orm
a) y = (x ndash 4)2 + 05
b) y = (x + 10)2 ndash 3
c) y = 8(x + 2)2 + 27
d) y = ndash32(x ndash 4)2 ndash 08
8 Determine the y -intercept or
y = (7 ndash 3x )(2x ndash 6)
Review Chapters 4 to 6
60 MHR bull Chapter 6
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and
7212019 Geometry in Design
httpslidepdfcomreaderfullgeometry-in-design 6060
9 Factor each trinomial
a) x 2 + 11x + 24 b) x 2 + x ndash 30 c) x 2 ndash 8x + 7 d) x 2 + 8x + 16
e) 3x 2 + 39x + 108 f) ndash10x 2 ndash 110x ndash 100
10 Which pairs contain equivalent
expressions How do you know
a) x 2 + 4x ndash 5 (5 + x )(x ndash 1)
b) x 2 + 35x + 3 (x ndash 2)(x ndash 15)
c) x 2 + x ndash 2 (x ndash 1)(x + 2)
11 Find the zeros by actoring Check by
substituting your answers or x
a) y = x 2 ndash x ndash 20
b) y = 10x 2 ndash 360
c) y = ndash2x 2 + 4x + 70
d) y = 35x 2 + 21x + 315
12 Te height o a rectangular prism is
16 cm more than the length o the base
Te width o the base i hal the length
a) Sketch and label a diagram to
represent the rectangular prism
b) Write an expression or the surace
area o the prism
c) What is the surace area o the prism i
the width o the base is 5 cm
d) What are the dimensions o the prism
i the surace area is 640 cm
2
14 a) Use linking cubes to model the capit
letter ldquoLrdquo Te base o the L should be10 cm and the height should be 20 cm
Te letter should be 10 cm thick Let
the side o a cube represent 5 cm
b) Use the model to help you draw an
isometric perspective drawing
c) Use the model to help you sketch a s
o orthographic drawings
15 A car dealership has ordered a platorm
to display their newest car models Anisometric perspective drawing or the
display platorm is shown Te distance
between pairs o vertical or horizontal
dots represents 2 m
a) Draw a set o orthographic drawings
or the platorm
b) Draw a net that can be used to make
a scale model or the platorm Te
rectangles that will become the slope
part o the platorm should have a
length o 3 units and a width o
225 units Cut out the net and