Understand the idea.
Same shape, possibly different size
Similar shapes have equal corresponding angles and one common multiplier for all corresponding lengths. Define the direction of the scale factor first, such as . A factor above 1 enlarges; a positive factor below 1 reduces.
In the diagram, becomes , becomes and becomes . Each ratio is 2. Equal differences alone would not show similarity; the same multiplier must work throughout.
Build one reliable equation
Find the scale factor from a known pair of corresponding sides, then multiply another original length by that factor. To return from the image to the original, divide by the factor. Keep the chosen direction visible in every ratio.
Convert lengths to the same unit before dividing. For instance, , so enlarging to has factor . The factor changes lengths while corresponding angles remain equal.
See the method in action.
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.
Worked solution
- 1
Find the factor from the smaller figure to the larger.
The known 6-centimetre side corresponds to the 9-centimetre side.
- 2
Apply that factor to the other smaller side.
Every corresponding length changes by the same factor in similar figures.
- 3
Calculate the larger length.
A factor greater than one must produce a length greater than 12 centimetres.
18 cm.
A reduction
A photograph is reduced from width 20 cm to width 8 cm. Its original height is 15 cm. Find the new height.
Worked solution
- 1
Calculate the new-to-original width ratio.
The reduced width is 40 percent of the original width.
- 2
Use the same ratio for height.
A photograph keeps its shape only if width and height share one scale factor.
- 3
Check that the proportions agree.
Both dimensions have the same new-to-original ratio.
6 cm.
Try it yourself.
Check your understanding
An enlargement changes a 5 cm side to 15 cm. What is its scale factor?
Show answer & reasoning
3.
The enlargement factor is the new length divided by the original: . Multiplying the original 5-centimetre side by 3 recovers 15 centimetres.
Keep practising
6 worksheet questions · 14 marks
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 . 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
.
2.5. - A similar enlargement has factor 3. A side is originally 7 cm. Find its image length.2 marks
cm.
21 cm. - A shape is reduced by factor . Its original side is 18 mm. Find the new length.2 marks
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 .
Height is 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.
.
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°.
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 ↗