Constructions

Scales

Scales are important in everyday life. We use scales to draw maps, to

construct building plans, in housing, street construction.... It is impossible

to draw building plans with the doors and window to size so we use scale

drawings.

If you are making orange squash and you mix one-part orange to four parts water, then the

ratio of orange to water will be 1:4 (1 to 4).

If you use 1 litre of orange, you will use 4 litres of water (1:4).

If you use 2 litres of orange, you will use 8 litres of water (2:8).

If you use 10 litres of orange, you will use 40 litres of water (10:40).

These ratios are all equivalent

1:4 = 2:8 = 10:40

Both sides of the ratio can be multiplied or divided by the same number to give an equivalent

ratio.

On a scale drawing, all dimensions have been reduced by the same proportion.

Example

A model boat is made to a scale of 1:20 (1 to 20). This scale can be applied to

any units, so 1mm measured on the model is 20mm on the actual boat, 1cm

measured on the model is 20cm on the actual boat, and so on.

a) If the 1:20 model boat is 15cm wide, how wide is the actual boat?

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b) If the boat has a mast of height 4m, how high is the mast on the model?

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Learning Intention: By the end of the lesson

you will be able to

Understand and use a scale to interpret a

plan

Luigi is inspecting a house plan where 1 cm:1 m, below there is a scale floor plan of his new

house.

Use a ruler to measure each length of each room and using the scale ratio, give the actual

lengths of the rooms in this house.

Writing Map Ratios

when writing may ratios it is important that you first turn the numbers into the same unit.

Then you can write the two numbers as a ratio or a fraction and use it to figure out the

length, width and height of the maps.

Example 3cm on a map gives a room’s height of 4m.

Map: Room

3: 400

3/400

Write down the map ratio and fraction of the following situations:

1. A map scale 2cm to 300m.

2. A distance of 3mm on the map represents 500m on the ground.

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3. A distance of 8km is represented by 4cm on the map.

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4. The scale of a map is 1 : 400, 000. If the distance between the two towns on the map is

3.8cm, find the actual distance between them. Give your answer in km.

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Constructing a triangle with three given sides.

Draw a triangle ABC of sides 8cm, 6cm and find the length of the third side.

First draw a rough sketch.

What is the length of

the third side?

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Learning Intention: By the end of the lesson

you will be able to

Make accurate drawings of

triangles

Draw a triangle of sides 7cm, 6cm and 5cm in the space below:

Constructing a triangle given 1 side and 2 angles. (ASA)

Construct triangle ABC, where AB = 11 cm, angle A = 35°, angle B = 42°.

Constructing a triangle given 2 sides and 1 angle. (SAS)

Construct triangle PQR where PQ = 12 cm, PR = 7 cm and angle P = 35°

On the given line below, construct triangle ABC such that AB = 8 cm, BC = 8 cm and angle B =

1000. Label your diagram.

Drawing a simple scale drawing

The ratio of this triangle is 1:3cm. Draw a drawing to scale in the following space:

1

3

2 Learning Intention: By

the end of the lesson

you will be able to

Draw simple scale

drawings.

The ratio of this triangle is 1:2. Draw the following triangle and find the third side of the

triangle.

2

3

Angle of Elevation and Depression

The image below is a model of Aya, point A, looking up looking up to Super Girl, point S, in

the sky.

What is the angle of elevation from Aya to Super Girl? ________

What is the angle of depression from Super girl to Aya? _______

When you see an object above you, there's an angle of elevation between the horizontal and your line of sight to the object.

when you see an object below you, there's an angle of depression between the horizontal and your line of sight to the object.

Learning Intention: By

the end of the lesson you

will be able to solve

problems involving

angles of elevation and

depression.

Write down whether the angle marked with a letter in each picture is an angle of

elevation or an angle of depression.

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iii)

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The figure below shows a tree and its shadow on the

ground. Let us draw a scale diagram using the given

information and hence find the height of the tree. Let 1

cm in the scale diagram represent an actual distance of 1

m.

From a sixty meter tall lighthouse a boat A is observed

at sea with an angle of depression of 31o and another

boat B with an angle of depression of 45o (see the

figure). The two boats and the lighthouse are in the

same vertical plane.

Draw a scale diagram with a scale of 1cm : 10m

depicting the above information and find the distance

between the boats A and B.

In a horizontal playground, Dilini is standing at the location A, 5 m away from a netball goal

post. She can see the top of the goal post T, with an angle of elevation of 18o from her eye

level E. She can see the base of the goal post F, from the same position with an angle of

depression of 15o. Draw a scale diagram and find Dilini’s height and the height of the goal

post.

When a diagram is not given, it is best to draw a sketch diagram prior to drawing the scale

diagram.

Sketch diagram:

Further examples

1) A person observes a rocket from a point 400 m horizontally away

from the launching pad when the rocket has travelled 700 m vertically

up from the launching pad. Using a scale diagram, find the angle of

elevation of the rocket.

2) A ladder leaning against a wall is shown in the figure. Draw a

scale diagram using the given information and find

(i) the length of the ladder and

(ii) the distance from the foot of the ladder to the wall.

3) A ramp for the use of wheelchairs to

access a building is shown in the figure.

Draw a suitable diagram using the given

information and find the length of the

ramp.

4) The top, P, of a clock tower situated on a horizontal ground

has an angle of elevation of 60o from a point A at the brink of a

building. The angle of elevation of P from a point B in the

building which is 5m directly above the point A, is 45o (see figure

on the right). Using a suitable scale diagram, find the height of

the clock tower and the distance from A to the foot Q of the

clock tower.

Constructing a perpendicular at a point on the line

1. Begin with line k, containing point P.

2. Place the compass on point P. Using an arbitrary radius,

draw arcs intersecting line k at two points. Label the

intersection points X and Y.

3. Place the compass at point X. Adjust the compass radius so

that it is more than (1/2)XY. Draw an arc as shown here.

4. Without changing the compass radius, place the compass

on point Y. Draw an arc intersecting the previously drawn arc.

Label the intersection point A.

5. Use the straightedge to draw line AP. Line AP is

perpendicular to line k.

Learning Intention: By

the end of the lesson

you will be able to

construct different

diagrams.

Constructing a perpendicular from a point to a line

1. Begin with point line k and point R, not on the line.

2. Place the compass on point R. Using an arbitrary radius,

draw arcs intersecting line k at two points. Label the

intersection points X and Y.

3. Place the compass at point X. Adjust the compass radius

so that it is more than (1/2)XY. Draw an arc as shown here.

4. Without changing the compass radius, place the compass

on point Y. Draw an arc intersecting the previously drawn

arc. Label the intersection point B.

5. Use the ruler to draw line RB. Line RB is perpendicular to

line k.

Construct the perpendicular bisector of a line segment, or construct the

midpoint of a line segment.

1. Begin with line segment XY.

2. Place the compass at point X. Adjust the compass radius

so that it is more than (1/2)XY. Draw two arcs as shown

here.

3. Without changing the compass radius, place the compass

on point Y. Draw two arcs intersecting the previously drawn

arcs. Label the intersection points A and B.

4. Using the ruler, draw line AB. Label the intersection

point M. Point M is the midpoint of line segment XY, and

line AB is perpendicular to line segment XY.

Construct the bisector of an angle.

1. Let point P be the vertex of the angle. Place the

compass on point P and draw an arc across both

sides of the angle. Label the intersection points Q

and R.

2. Place the compass on point Q and draw an arc

across the interior of the angle.

3. Without changing the radius of the compass,

place it on point R and draw an arc intersecting the

one drawn in the previous step. Label the

intersection point W.

4. Using the ruler, draw line PW. This is the bisector

of QPR.

Constructing a right angle (90o)

1). Use ruler and draw a Line segment

OB of any convenient length. (as shown

below)

2). Now use compass and open it to any

convenient radius. And with O as center ,

draw an arc which cuts line segment OB

at X.

3). Again use compass and opened to the

same radius (as of step 2). And with X as

center , draw an arc which cuts first arc

at D . (as shown below)

4). Again use compass and opened to the

same radius (as of step 2). And with D as

center, draw another arc which cuts first

arc at C . (as shown below)

5) Again use compass and opened to the

same radius (as of step 2). And With C &

D as center , draw two arc which cuts

each other at E .

6) Join OE and extent it to A.

Construct an angle of 90o

Constructing a square or a rectangle

We start with a given line segment

AB> This will become one side of

the square.

Note. Steps 1 through 5 construct a perpendicular to line AB at the point B. This is the

same construction as Constructing the perpendicular at a point on a line

1. Extend the line AB to the right.

2. Set the compasses on B and any

convenient width. Scribe an arc on

each side of B, creating the two

points F and G.

3. With the compasses on G and

any convenient width, draw an arc

above the point B.

4. Without changing the

compasses' width, place the

compasses on F and draw an arc

above B, crossing the previous arc,

and creating point H

5. Draw a line from B through H.

This line is perpendicular to AB, so

the angle ABH is a right angle

(90°);

This will become the second side

of the square

We now create four sides of the square the same length as AB

6. Set the compasses on A and set

its width to AB. This width will be

held unchanged as we create the

square's other three sides.

7. Draw an arc above point A.

8. Without changing the width,

move the compasses to point B.

Draw an arc across BH creating

point C - a vertex of the square.

9. Without changing the width,

move the compasses to C. Draw an

arc to the left of C across the

exiting arc, creating point D - a

vertex of the square.

10. Draw the lines CD and AD

Construct a rectangle ABCD in which AB = 5 cm

and BC = 4 cm.

Drawing Diagonals

To draw a diagonal of a shape, you must bisect the angle as

shown before.

Example: Draw the following rectangle in the scale of 1: 2 and

draw the diagonals. How long are the diagonals?

3

1.5

Further examples

1) Construct the following triangles

2)

Construct the triangle KLM that is right-angled at M, with KM= 6cm and KL= 10.5cm.

Measure and write down the length of side LM and the two acute angles K and L.

3)

4)

5) Construct rectangle ABCD with AB = 8.4cm and diagonal AC= 8.9cm. Hence measure the

height BC and calculate the area of ABCD.

8.4cm

8.9cm