Similar shapes and corresponding sides | Year 8

01 / LEARN

The idea, step by step

Same shape, possibly different size

Similar shapes have equal corresponding angles and proportional corresponding lengths. All length ratios must use the same direction: new divided by original, for example. A scale factor greater than 1 enlarges; a scale factor between 0 and 1 reduces.

In the diagram, each side of the larger triangle is twice the matching side of the smaller. The same factor applies to 3, 4 and 5 cm, producing 6, 8 and 10 cm.

Build one reliable equation

Choose a pair of known corresponding lengths to obtain the scale factor. Apply that factor to another length in the original shape. If working in the reverse direction, divide instead.

Keep units consistent before taking a ratio. A side of 2 m matching a side of 50 cm gives factor $200 / 50 = 4$ from the smaller to the larger, not $2 / 50$ . Angles do not multiply by the scale factor.

A common trap

Using the same numerical difference for every pair of corresponding sides.

Instead: Similarity uses multiplication by one factor, not addition of one difference.

02 / WORKED EXAMPLES

See the thinking

Example 1

An enlargement

Two similar figures have corresponding sides 6 cm and 9 cm. Another side of the smaller figure is 12 cm. Find its match.

The factor from small to large is $9 / 6 = 1.5$ .
Apply the same factor: $12 (1.5) = 18$ cm.

18 cm.

Example 2

A reduction

A photograph is reduced from width 20 cm to width 8 cm. Its original height is 15 cm. Find the new height.

The reduction factor is $8 / 20 = 0.4$ .
New height is $15 (0.4) = 6$ cm.
Both dimensions have changed by the same factor, so the image keeps its shape.

6 cm.

Check your understanding

An enlargement changes a 5 cm side to 15 cm. What is its scale factor?

Show answer

The new-to-original ratio is $15 / 5 = 3$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A 4 cm side becomes 10 cm in a similar shape. Find the scale factor.
2 marks
A similar enlargement has factor 3. A side is originally 7 cm. Find its image length.
2 marks
A shape is reduced by factor $\frac{1}{2}$ . Its original side is 18 mm. Find the new length.
2 marks
Similar rectangles have widths 8 cm and 12 cm. The smaller has height 10 cm. Find the larger height.
3 marks
A 30 cm model side represents an actual side of 1.2 m. Find the scale factor from model to actual.
3 marks
A triangle is enlarged by factor 4. One original angle is 50°. What is the corresponding image angle? Explain.
2 marks

Read the worked solutions

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

A 4 cm side becomes 10 cm in a similar shape. Find the scale factor.
2 marks
$k = 10 / 4 = 2.5$ .
2.5.
A similar enlargement has factor 3. A side is originally 7 cm. Find its image length.
2 marks
$7 (3) = 21$ cm.
21 cm.
A shape is reduced by factor $\frac{1}{2}$ . Its original side is 18 mm. Find the new length.
2 marks
$18 (1 / 2) = 9$ mm.
9 mm.
Similar rectangles have widths 8 cm and 12 cm. The smaller has height 10 cm. Find the larger height.
3 marks
Factor is $12 / 8 = 1.5$ .
Height is $10 (1.5) = 15$ cm.
15 cm.
A 30 cm model side represents an actual side of 1.2 m. Find the scale factor from model to actual.
3 marks
Convert 1.2 m to 120 cm.
$k = 120 / 30 = 4$ .
4.
A triangle is enlarged by factor 4. One original angle is 50°. What is the corresponding image angle? Explain.
2 marks
Similarity preserves corresponding angles; only lengths are scaled.
50°.

Checking saved progress…

Continue learning

Triangle similarity

Next lesson

Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Congruent shapes and matching parts; triangle congruence tests; similar shapes and corresponding sides; triangle similarity; properties of quadrilaterals established using triangles; testing congruence/similarity with transformations; decision-tree algorithms and digital geometry.

Official curriculum source ↗

Back to Congruence and similarity