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First teaching 2023

First exams 2025

Similarity (CIE IGCSE Maths: Core)

Revision Note

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Author

Naomi C

Last updated

15 February 2024

Similarity

What are similar shapes?

Two shapes are similar if they have the same shape and their corresponding sides are in proportion
- One shape is an enlargement of the other

How do we prove that two triangles are similar?

To show that two triangles are similar you need to show that their angles are the same
- If the angles are the same then corresponding lengths of a triangle will automatically be in proportion
You can use angle properties to identify equal angles
- Look out for for isosceles triangles, vertically opposite angles and angles on parallel lines
If a question asks you to prove two triangles are similar
- For each pair of corresponding angles
  - State that they are of equal size
  - Give a reason for why they are equal

How do we prove that two shapes are similar?

To show that two non-triangular shapes are similar you need to show that their corresponding sides are in proportion
- Divide the length of one side by the length of the corresponding side on the other shape to find the scale factor
If the scale factor is the same for all corresponding sides, then the shapes are similar

Examiner Tip

A pair of similar triangles can often be opposite each other in an hourglass formation.
- Look out for the vertically opposite, equal angles.
- It may be helpful to sketch the triangles next to each other and facing in the same direction.

Worked example

(a)

Prove that the two rectangles shown in the diagram below are similar.

Two similar rectangles

(a)

Use the corresponding lengths (15 cm and 6 cm) to find the scale factor

$\frac{15}{6} = 2.5$

Use the corresponding width (5 cm and 2 cm) to find the scale factor for the other pair of sides

$\frac{5}{2} = 2.5$

The two rectangles are similar, with a scale factor of 2.5

(b)

In the diagram below, AB and CD are parallel lines.
Show that triangles ABX and CDX are similar.

Two similar triangles

State the equal angles by name, along with clear reasons
Don't forget to state that similar triangles need to have equal corresponding angles

Angle AXB = angle CXD (vertically opposite angles are equal)
Angle ABC = angle BCD (alternate angles on parallel lines are equal)
Angle BAD = angle ADC (alternate angles on parallel lines are equal)

All three corresponding angles are equal, so the two triangles are similar

Similar Lengths

How do I solve problems that involve similar lengths?

Equivalent lengths in two similar shapes will be in the same ratio and are linked by a scale factor
- Identify known lengths of corresponding sides
- Establish the type of enlargement
  - If the shape is getting bigger, then the scale factor is greater than 1
  - If the shape is getting smaller, then the scale factor is greater than 0 but less than 1
- Find the scale factor
  - Divide a known length on the second shape by the corresponding known length on the first shape
- Use the scale factor to find the length you need
  - Multiply a known length by the scale factor on the first shape to find the corresponding length on the second shape
  - Divide a known length on the second shape by the scale factor to find the corresponding length on the first shape

Examiner Tip

If similar shapes overlap on the diagram (or are not clear) draw them separately.
- For example, in this diagram the triangles ABC and APQ are similar:
- So redraw them separately before starting:

Worked example

ABCD and PQRS are similar shapes.
Two similar quadrilaterals Find the length of PS.

The two shapes are mathematically similar
Each length on the first shape can be multiplied by a scale factor to find the corresponding length on the second shape

Identify two known corresponding sides of the similar shapes

AB and PQ are corresponding sides

The second shape is smaller than the first shape so the scale factor will be between 0 and 1
Divide the known length on the second shape by the corresponding length on the first shape to find the scale factor

$\begin{array}{rcl} Scale Factor & = & \frac{3}{6} = \frac{1}{2} \end{array}$

Multiply the length AD by the scale factor to find its corresponding length PS on the second shape

$\begin{array}{rcl} P S & = & \frac{1}{2} \times 15 = \frac{15}{2} \end{array}$

$P S = 7.5 cm$

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