12 CONSTRUCTIONS AND LOCI - · PDF file12 CONSTRUCTIONS AND LOCI ... 4cm 6cm C B AB A C B...

12 CONSTRUCTIONS AND LOCI

In this chapter you will: use a ruler and a pair of compasses to draw

triangles given the lengths of the sides use a straight edge and a pair of compasses to

construct perpendiculars and bisectors construct and bisect angles using a pair of

compasses draw loci and regions learn how to draw, use and interpret scale

drawings.

Architects make scale drawings of projects they are working on for both planning and presentation purposes. Originally these were done on paper using ink, and copies had to be made laboriously by hand. Later they were done on tracing paper so that copying was easier. Computer-generated drawings have now largely taken over, but, for many of the top architecture fi rms, these too have been replaced, by architectural animation.

Objectives Before you start

You need to: be able to make accurate drawings of triangles

and 2D shapes using a ruler and a protractor be able to draw parallel lines using a protractor

and ruler have some understanding of ratio be able to change from one metric unit of length

to another.

194

195

12.1 Constructing triangles

12.1 Constructing triangles

You can draw a triangle when given the lengths of its sides.

Objective

If you were redesigning a garden and wanted a triangular border you would need to make a plan fi rst and draw the triangles accurately.

Why do this?

1. Use a ruler and protractor to make an accurate drawing of this triangle. Measure AC, BC and angle ACB.

Get Ready

Two triangles are congruent if they have exactly the same shape and size. One of four conditions must be true for two triangles to be congruent: SSS, SAS, ASA and RHS (see Section 8.1).

Constructing a triangle using any one of these sets of information therefore creates a unique triangle. More than one possible triangle can be created from other sets of information.

Key Points

Make an accurate drawing of the triangle shown in the sketch.Example 1

Start by drawing the base 5 cm long. Label the ends A and B.

Draw an arc, radius 4 cm. centre A.Draw an arc, radius 6 cm, centre B.

C is the point where the two arcs intersect.C is 4 cm from A and 6 cm from B.Join C to A and C to B.

A 5 cm

6 cm4 cm

C

B

A B

A

C

B

Watch Out!

The diagram in the question will not be drawn accurately so don’t measure it, use the dimensions marked.

Watch Out!

60° 41°9 cmA B

C

Chapter 12 Constructions and loci

196

Show that there are two possible triangles ABC in which AB 5.6 cm, BC 3.3 cm and angle A 31°.

Example 2

BA

B31°

A

31°BA

C1

C2

Draw the line AB with length 5.6 cm.

Using a protractor, draw an angle of 31° at A.

Draw an arc of 3.3 cm from point B, to locate the possible positions of C.Triangle ABC1 and ABC2 both have the given measurements.

1 Here is a sketch of triangle XYZ.

4.5 cmY

Z

X

7.5 cm6 cm

Construct triangle XYZ.

2 Construct an equilateral triangle with sides of length 5 cm.

3 Construct the triangle XYZ with sides XY 4.2 cm, YZ 5.8 cm and ZX 7.5 cm.

4 Here is a sketch of the quadrilateral CDEF.

C 6 cm

4.5 cm

3.5 cm

4 cm 5 cm

E

F

D

Make an accurate drawing of quadrilateral CDEF.

5 The rhombus KLMN has sides of length 5 cm.The diagonal KM 6 cm.Make an accurate drawing of the rhombus KLMN.

6 Explain why it is not possible to construct a triangle with sides of length 4 cm, 3 cm and 8 cm.

Exercise 12A

D

Questions in this chapter are targeted at the grades indicated.

197

12.2 Perpendicular lines

bisector perpendicular bisector construction line segment

12.2 Perpendicular lines

You can construct perpendicular lines using a straight edge and compasses.

Objective

Many structures involve lines or planes that are perpendicular, for example the walls and fl oor of a house are perpendicular.

Why do this?

1. Draw a circle with a radius of 4 cm.2. Mark two points A and B 6 cm apart. Mark the points that are 5 cm from A and 5 cm from B.3. Draw two straight lines which are perpendicular to each other.

Get Ready

A bisector cuts something exactly in half. A perpendicular bisector is at right angles to the line it is cutting. You can use a straight edge and compass in the construction of the following:

the perpendicular bisector of a line segment the perpendicular to a line segment from a point on it the perpendicular to a line segment from a point not on the line.

Key Points

Construct the perpendicular bisector of the line AB.

Construct the perpendicular to a point P on a line AB.

Example 3

Draw arcs centred on A above and below the line AB with a radius more than half of AB.

Draw arcs centred on B above and below the line AB with the same radius as before.

Join C to D. CD is the required perpendicular bisector.

Example 4

Join C to D. CD is the required perpendicular bisector.

Draw arcs with the same radius centre P to cut AB at X and Y.

Construct the perpendicular bisector of XY. Since PY PX, this must go through P.

A B

A B

C

D

A BYPX

A BYPX


198

Construct the perpendicular to a line AB from a point P not on the line.Example 5

Start by drawing arcs with the same radius, centre P to cut the line (extended if necessary) at X and Y.

Then construct the perpendicular bisector of XY.

A BY

P

X

A BY

P

X

1 Draw line segments of length 10 cm and 8 cm. Using a straight edge and a pair of compasses, construct the perpendicular bisector of each of these line segments.

2 Draw these lines accurately, and then construct the perpendicular from the point P.a

A

B

P 7 cm3 cm

b A

B

P9 cm

2 cm

3 Draw a line segment AB, a point above it, P, and a point below it, Q. Construct the perpendicular from P to AB, and from Q to AB.

Exercise 12B

C

12.3 Constructing and bisecting angles

You can construct certain angles using compasses. You can construct the bisector of an angle using a

straight edge and compasses. You can construct a regular hexagon inside a circle.

Objectives

You may need to bisect an angle accurately when cutting a tile to place in an awkward corner.

Why do this?

1. Draw a circle with a radius of 3 cm.2. Draw an angle of 60°.3. Use a protractor to bisect an angle of 60°.

Get Ready

199angle bisector

12.3 Constructing and bisecting angles

Construct an angle of a 60° b 120°.

a

b

Construct the bisector of the angle ABC.

Example 6

Example 7

A B

A B

A60°

B

A B

120°

CD

Start by drawing an arc at the point A. Where the arc cuts the line, label the point B.

Keep your compasses the same width and put the point at B. Draw an arc to cut the fi rst one.

Join up to get a 60º angle. Label the point C.ABC is an equilateral triangle.

Draw a longer fi rst arc and then draw a third arc from point C with the same radius.

Start by drawing arcs with the same radius (or a single arc), centre B to cut the arms BA and BC at X and Y.

Then draw an arc centre X radius BX and an arc centre Y radius BX to cross at D.

Join D to B to get the angle bisector.

B C

A

B Y C

A

X

B Y C

A

DX


200

1 Copy the diagrams and construct the bisector of the angle ABC.a

B

A

C

b B

A

C

Exercise 12C

Construct a regular hexagon inside a circle.Example 8

A

Draw a circle and mark a point A on its circumference.

A

B

A

B

Keep the compasses set at the size of the radius, and from point A draw an arc that cuts the circle at point B.

Repeat the process until six points are marked on the circumference. Join the points to make a regular hexagon.

C

201

12.4 Loci

locus equidistant

2 Copy the diagrams and construct the bisector of angle Q in the triangle PQR.a R

Q P

b Q

P R

3 Construct each of the following angles.a 60° b 120° c 90° d 30° e 45°

4 Draw a regular hexagon in a circle of radius 4 cm.

5 Draw a regular octagon in a circle of radius 4 cm.

12.4 Loci

You can draw the locus of a point.

Objective

Scientists studying interference effects of radio waves need to plot paths that are equidistant from two or more transmitters. They use loci to do this.

Why do this?

1. Put a cross in your book. Mark some points which are 3 cm from the cross.2. Put two crosses A and B less than 3 cm apart in your book. Mark points which are 3 cm from each cross.3. Draw two parallel lines. Mark any points which are the same distance from both lines.

Get Ready

A locus is a line or curve, formed by points that all satisfy a certain condition. A locus can be drawn such that:

its distance from a fi xed point is constant it is equidistant from two given points its distance from a given line is constant it is equidistant from two lines.

Key Points

C

Show the locus of all points which are at a distance of 3 cm from the fi xed point O.

Example 9

The locus is a circle, radius 3 cm, centre O.All the points on the circle are 3 cm from O. O

3 cm


202

Show the locus of all points which are equidistant from the points X and Y.

Show the locus of all points which are 3 cm from the line segment XY.

Example 10

Example 11

Construct the perpendicular bisector of the line XY.

All points on the perpendicular bisector are equidistant (the same distance) from X and Y.

X Y

Examiner’s Tip

Draw loci accurately. Use a pair of compasses, a ruler and a straight edge.

All points on the semicircles are 3 cm from the point Y.

X Y

3 cm These lines are parallel to XY and 3 cm away from it.

1 Mark two points A and B approximately 6 cm apart. Draw the locus of all points that are equidistant from A and B.

2 Draw the locus of all points which are 3.5 cm from a point P.

3 Draw the locus of a point that moves so that it is always 1.5 cm from a line 5 cm long.

4 Draw two lines PQ and QR, so that the angle PQR is acute. Draw the locus of all points that are equidistant between the two lines PQ and QR.

Exercise 12D

C

203

12.5 Regions

region

12.5 Regions

You can draw regions.

Objective

If you tether a goat to a point in your garden to eat the grass, you might want to check that the region it can access doesn’t include the fl owerbed.

Why do this?

1. Put a cross in your book. Mark some points which are less than 3 cm from the cross.2. Put two crosses A and B in your book. Mark points which are closer to A than to B.3. Draw two parallel lines. Mark any points which are further from one line than the other.

Get Ready

A set of points can lie inside a region rather than on a line or curve.

The region of points can be drawn such that: the points are greater than or less than a given distance from a fi xed point the points are closer to one given point than to another given point the points are closer to one given line than to another given line.

Key Points

Draw the region of points which are less than 2 cm from the point O.

O

Draw the region of all points which are closer to the point X than to the point Y.

X Y

Example 12

Example 13

The locus is a circle, radius 2 cm, centre O.All the points on the circle are 2 cm from O.

All the points to the left of the perpendicular bisector of XY are closer to X than to Y.


204

ABCD is a square of side 4 cm. Draw the region of points inside the rectangle that are both more than 3 cm from point A and more than 2 cm from the line BC.

Example 14C

4 cm

4 cm

D

BAC

3 cm

D

BA

C

2 cm

D

BA

CD

BA

Find the locus of points 3 cm from point A inside the square.

Find the locus of points 2 cm from the line BC inside the square.

Shade the area that is both more than 3 cm from point A and more than 2 cm from the line BC.

1 Shade the region of points which are less than 2 cm from a point P.

2 Shade the region of points which are less than 2.6 cm from a line 4 cm long.

3 Mark two points, G and H, roughly 3 cm apart. Shade the region of points which are closer to G than to H.

4 Draw two lines DE and EF, so that the angle DEF is acute. Shade the region of points which are closer to EF than to DE.

5 Baby Tommy is placed inside a rectangular playpen measuring 1.4 m by 0.8 m. He can reach 25 cm outside the playpen. Show the region of points Tommy can reach beyond the edge of the playpen.

Exercise 12E

C

205

12.6 Scale drawings and maps

scale factor scale diagram


You can read and construct scale drawings. You can draw lines and shapes to scale and

estimate lengths on scale drawings. You can work out lengths using a scale factor.

Objectives

When a new aeroplane is being designed or an extension to a house is planned, accurate scale drawings have to be made.

Why do this?

1. Convert from cm to km: 2. Convert from km to cm:a 5 000 000 cm b 250 000 cm. a 4 km b 0.3 km.

Get Ready

Here is a picture of a scale model of a Saturn rocket. The model has been built to a scale of 1 : 24. This means that every length on the model is shorter than

the length on the real rocket, with a length of 1 cm on the model representing a length of 24 cm on the real rocket.

The real rocket is an enlargement of the model with a scale factor of 24; the model is a smaller version of the real rocket with a scale factor of 1 __ 24 .

In general, a scale of 1 : n means that: a length on the real object the length on the scale diagram or model n a length on the scale drawing or model the length on the real object n.

Key Points

The Empire State Building is 443 m tall. Bill has a model of the building that is 88.6 cm tall.a Calculate the scale of the model. Give your answer in the form 1 : n.b The pinnacle at the top of Bill’s model is 12.4 cm in length. Work out the actual length of

the pinnacle at the top of the Empire State Building. Give your answer in metres.

a Height of building 443 100 44 300 cm

Scale factor 44 300 _______ 88.6

500

Scale of model 1 : 500

b Length of pinnacle on building 12.4 500 6200 cmLength of pinnacle on building 6200 100 62 m

Example 15

Both heights have to be in the same units.Change 443 m to cm by multiplying by 100.

Scale factor Height of building

_______________ Height of model

Length on model Length on building 500.Length on building Length on model 500.

Change cm to m by dividing by 100.


206

The scale of a map is 1 : 50 000. a On the map, the distance between two churches is 6 cm. Work out the real distance

between the churches. Give your answer in kilometres.b The real distance between two train stations is 12 km. Work out the distance between

the two train stations on the map. Give your answer in centimetres.

Method 1a Real distance between churches

6 50 000 300 000 cm 3000 m 3 km

b 12 km 12 1000 100 1 200 000 cmDistance between stations on map 1 200 000 50 000 24 cm

Method 2Map distance of 1 cm represents real distance of 0.5 km.a 6 cm on the map represents real

distance of 6 0.5 3 km.Distance between the churches 3 km.

b Real distance of 12 km represents map distance of 12 0.5 24 cm. Distance between the stations on map 24 cm.

Example 16

A scale of 1 : 50 000 means:real distance map distance 50 000.

Change cm to m, divide by 100.Change m to km, divide by 1000.

Change km to cm by multiplying by 1000 100.

Map distance real distance 50 000

1 : 50 000 means 1 cm : 50 000 cmor 1 cm : 500 mor 1 cm : 0.5 km

1 This is an accurate map of a desert island. There is treasure buried on the island at T. Key to mapP palm trees R rocksC cliffs T treasureThe real distance between the palm trees and the cliffs is 5 km.a Find the scale of the map. Give your answer in the form 1 cm represents n km, giving the value of n. b Find the real distance of the treasure from: i the cliffs ii the palm trees iii the rocks.

P

T

R

C

Exercise 12F

D AO2AO3

207

2 On a map of England, 1 cm represents 10 km. a The distance between Hull and Manchester is 135 km. Work out the distance between Hull and

Manchester on the map.b On the map, the distance between London and York is 31.2 cm. Work out the real distance between

London and York.

3 Here is part of a map, not accurately drawn, showing three

A 8 km

12 km

B

C

N towns: Alphaville (A), Beecombe (B) and Ceeton (C).a Using a scale of 1 : 200 000, accurately draw this part of

the map. b Find the real distance, in km, between Beecombe and

Ceeton. c Use the scaled drawing to measure the bearing of

Ceeton from Beecombe.

4 This is a sketch of Arfan’s bedroom. It is not drawn to scale. 4 m

3 m

3 m

1.5 m

1 m

2.5 m

Draw an accurate scale drawing on cm squared paper of Arfan’s bedroom. Use a scale of 1 : 50.

5 A space shuttle has a length of 24 m. A model of the space shuttle has a length of 48 cm.a Find, in the form 1 : n, the scale of the model.b The height of the space shuttle is 5 m. Work out the height of the model.

6 The distance between Bristol and Hull is 330 km. On a map, the distance between Bristol and Hull is 6.6 cm.a Find, as a ratio, the scale of the map. b The distance between Bristol and London is 183 km. Work out the distance between Bristol and

London on the map. Give your answer in centimetres.


D

C

AO2

Two triangles are congruent if they have exactly the same shape and size. One of four conditions must be true for two triangles to be congruent: SSS, SAS, ASA and RHS.

Constructing a triangle using any one of these sets of information therefore creates a unique triangle. More than one possible triangle can be created from other sets of information. A bisector cuts something exactly in half. A perpendicular bisector is at right angles to the line it is cutting. A locus is a line or curve, formed by points that all satisfy a certain condition. A locus can be drawn such that

its distance from a fi xed point is constant it is equidistant from two given points its distance from a given line is constant it is equidistant from two lines.

Chapter review


208

A set of points can lie inside a region rather than on a line or curve. A region of points can be drawn such that:

the points are greater than or less than a given distance from a fi xed point the points are closer to one given point than to another given point the points are closer to one given line than to another given line.

A scale of 1 : n means that: a length on the real object the length on the scale diagram or model n a length on the scale drawing or model the length on the real object n.

Review exercise

1 AB 8 cm. AC 6 cm. Angle A 52°. C

A 8 cm

6 cm

B

52°

Diagram NOTaccurately drawn

Make an accurate drawing of triangle ABC.

Nov 2008

2 Make an accurate drawing of the

A

D

C

8 cm

5 cm

120° 4 cm

B


quadrilateral ABCD.

3 Make an accurate drawing of triangle ABC.

A 6.5 cm

C

B

60° 30°


May 2009

4 Make an accurate drawing of triangle PQR.

Q

P

8.7 cm

13.9 cm7.3 cm

R


5 A model of the Eiffel Tower is made to a scale of 2 millimetres to 1 metre. The width of the base of the real Eiffel Tower is 125 metres.

a Work out the width of the base of the model. Give your answer in millimetres. The height of the model is 648 millimetres.

b Work out the height of the real Eiffel Tower. Give your answer in metres June 2008, adapted

D

209

Chapter review

6 Beeham is 10 km from Alston.Corting is 20 km from Beeham.Deetown is 45 km from Alston.The diagram below shows the straight road from Alston to Deetown.This diagram has been drawn accurately using a scale of 1 cm to represent 5 km.

Alston Deetown

On a copy of the diagram, mark accurately with crosses (x), the positions of Beeham and Corting.Nov 2007

7 ABC is a triangle.

B

A

C

Copy the triangle accurately and shade the region inside the triangle which is both less than 4 centimetres from the point B and closer to the line AC than the line AB.

June 2009, adapted

8 On a copy of the diagram, use a ruler and pair of compasses to construct an angle of 30° at P.You must show all your construction lines.

P

Nov 2007, adapted

9 a Mark the points C and D approximately 8 cm apart. Draw the locus of all points that are equidistantfrom C and D.

d Draw the locus of a point that moves so that it is always 3 cm from a line 4.5 cm long.

10 B is 5 km north of A.

Diagram NOTaccurately drawn7 km5 km

4 kmC

B

A

N C is 4 km from B. C is 7 km from A. a Make an accurate scale drawing of triangle ABC. Use a scale of 1 cm to 1 km. b From your accurate scale drawing, measure the bearing of C from A. c Find the bearing of A from C. Nov 2000

Exam Question Report

79% of students answered this question poorly because they did not use two different constructions.

C


210

11 On an accurate copy of the diagram use a ruler and pair of compasses to construct the bisector of angle ABC.You must show all your construction lines. A

C

B

Nov 2008, adapted

12 ABCD is a rectangle. A B

D C

Make an accurate drawing of ABCD.Shade the set of points inside the rectangle which are both more than 1.2 centimetres from the point Aand more than 1 centimetre from the line DC.

13 Draw a line segment 7 cm long. Construct the perpendicular bisector of the line segment.

14 Draw a line segment ST and a point above it, M. Construct the perpendicular from M to ST.

15 As a bicycle moves along a fl at road, draw the locus of: a the yellow dotb the green dot.

16

Draw the locus of a man’s head as the ladder he is on slips down a wall.

C

AO3

AO3

AO3

12 CONSTRUCTIONS AND LOCI - · PDF file12 CONSTRUCTIONS AND LOCI ... 4cm 6cm C B AB A C B...

Documents

Transcript of 12 CONSTRUCTIONS AND LOCI - · PDF file12 CONSTRUCTIONS AND LOCI ... 4cm 6cm C B AB A C B...