Finding missing lengths by similarity | Year 9

01 / LEARN

Understand the idea.

Find the multiplier once

When similarity is established, find the scale factor using two known corresponding sides. Write image length over original length. Multiply another original length by that factor; divide if the required length belongs to the original figure.

Keep the side names beside the numbers until the correspondence is secure. The same multiplier must connect every matched pair. It cannot connect a shorter side in one triangle to the hypotenuse in the other merely because those lengths are known.

image length = k \times original length

Indirect measurements need geometric reasons

Shadow measurements can form similar right triangles when the objects are vertical, the ground is level and sunlight has the same direction. Equal right angles and equal sunlight angles justify comparing height with shadow length.

Measure the shadows at the same time and in consistent units. Sloping ground or a change in the Sun's angle can break the matching-angle argument. State these assumptions because the proportional calculation depends on them.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Find an image length

In the supplied similar triangles, find $x$ .

Worked solution

1
Identify the known corresponding pair.
$A B D E$
The stated similarity pairs the eight-centimetre side with the twelve-centimetre side.
2
Find the original-to-image scale factor.
$k = \frac{D E}{A B} = \frac{12}{8} = 1.5$
Every image length is one and a half times its corresponding original.
3
Scale AC to find DF.
$x = 6 \times 1.5 = 9 cm$
AC and DF correspond, so they use the same multiplier.

Answer

9 cm

EXAMPLE 02

Use shadows

At the same time on level ground, a 1.5 m pole casts a 2 m shadow. A tree casts a 12 m shadow. Find the tree height.

Worked solution

1
Justify the matching triangles.
The vertical objects on level ground form right angles, and simultaneous sunlight gives equal acute angles.
2
Compare the shadow lengths.
$k = \frac{12}{2} = 6$
The tree's triangle is six times the pole's triangle in corresponding lengths.
3
Scale the pole height.
$h = 1.5 \times 6 = 9 m$
Similar triangles have equal height-to-shadow ratios.
4
Check the two ratios.
$\frac{9}{12} = \frac{1.5}{2} = 0.75$
Both heights are three quarters of their shadow lengths under these conditions.

Answer

9 m

03 / YOUR TURN

Try it yourself.

Check your understanding

An original side is 7 cm. Its corresponding image side is 21 cm. Another original side is 5 cm. Find its image.

Show answer & reasoning

15 cm

The original-to-image factor is $\frac{21}{7} = 3$ . Apply that factor to the other original side: $5 \times 3 = 15 cm$ .

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6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Two similar triangles have corresponding sides 5 cm and 15 cm. Find the scale factor from the first to the second.
2 marks
With scale factor 3, an original side is 8 cm. Find its image length.
2 marks
A 28 cm image side was made using scale factor 4. Find its original length.
2 marks
Similar rectangles measure 8 cm by 5 cm and 20 cm by $w$ cm. Find $w$ .
2 marks
A 1.8 m person casts a 1.2 m shadow. A flagpole casts a 6 m shadow at the same time on level ground. Find its height.
3 marks
Similar triangles have perimeters 24 cm and 60 cm. A side of the larger is 22.5 cm. Find the corresponding smaller side.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Two similar triangles have corresponding sides 5 cm and 15 cm. Find the scale factor from the first to the second.
2 marks
$k = \frac{15}{5} = 3$ .
$3$
With scale factor 3, an original side is 8 cm. Find its image length.
2 marks
$8 \times 3 = 24$ cm.
$24$ cm
A 28 cm image side was made using scale factor 4. Find its original length.
2 marks
$28 \div 4 = 7$ cm.
$7$ cm
Similar rectangles measure 8 cm by 5 cm and 20 cm by $w$ cm. Find $w$ .
2 marks
$k = \frac{20}{8} = 2.5$ .
$w = 5 \times 2.5 = 12.5$ cm.
$12.5$ cm
A 1.8 m person casts a 1.2 m shadow. A flagpole casts a 6 m shadow at the same time on level ground. Find its height.
3 marks
Shadow scale factor $= \frac{6}{1.2} = 5$ .
Height $= 1.8 \times 5 = 9$ m.
$9$ m
Similar triangles have perimeters 24 cm and 60 cm. A side of the larger is 22.5 cm. Find the corresponding smaller side.
3 marks
Length factor is $\frac{60}{24} = 2.5$ .
Smaller side $= \frac{22.5}{2.5} = 9$ cm.
$9$ cm

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 proportional modelling, rates, scale, similarity and enlargement. Assumptions, interpretation and model limitations are taught alongside calculation.

Official curriculum source ↗

In this lesson

Understand the idea.

Find the multiplier once

Indirect measurements need geometric reasons

See the method in action.

Find an image length

Identify the known corresponding pair.

Find the original-to-image scale factor.

Scale AC to find DF.

Use shadows

Justify the matching triangles.

Compare the shadow lengths.

Scale the pole height.

Check the two ratios.

Try it yourself.

Check your understanding

Keep practising