# Similar triangles · Year 9

> Similar triangles, Year 9: Enlarging and reducing shapes and more. Explanation, worked examples and free practice for the national curriculum in England.

In Similar triangles (Year 9) you practise 4 skills: Enlarging and reducing shapes · Corresponding sides of similar triangles · Two equal angles: are the triangles similar? · Angles, perimeters and lengths in similar triangles.

Two shapes are similar if one is an enlargement or a reduction of the other. Matching angles are equal, and all the sides are multiplied by the same number: the scale factor.

When similar triangles are written down, the order of the letters shows which vertices match: the first vertex matches the first, the second the second, and the third the third.

By the AA rule, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. The third angle follows.

A line parallel to a side of a triangle cuts off a small triangle that is similar to the whole triangle. The perimeters of similar triangles are in the same ratio as the sides too.

**Tip:** To find the scale factor, divide a side of the new triangle by the matching side of the original triangle.

## What you learn

### Enlarging and reducing shapes

- Example: A rectangle 13 cm long and 2 cm wide is enlarged by a scale factor of 2. What is the length of the new rectangle?
- Answer: 26 cm
- Solution: The length of the original rectangle is 13 cm. Multiply by the scale factor: 13 × 2 = 26 The length of the new rectangle is 26 cm.

### Corresponding sides of similar triangles

- Example: The triangles are similar: ΔABC ∼ ΔDEF . The lengths are in centimetres. In the first triangle, AB = 12, BC = 6, AC = 10 . In the second triangle, EF = 12 . What is the length of side DE ?
- Answer: 24 cm
- Solution: Side EF matches side BC , so the scale factor is 12 ÷ 6 = 2 . The side you want matches side AB : DE = 12 × 2 = 24 . The length of the side is 24 cm.

### Two equal angles: are the triangles similar?

- Example: In triangle ABC : ∢A = 30°, ∢B = 115° . In triangle DEF : ∢E = 35°, ∢F = 115° . Are the triangles similar?
- Options: Yes · No
- Answer: Yes
- Solution: Work out the third angle in each triangle: ∢C = 180° − 30° − 115° = 35° ∢D = 180° − 35° − 115° = 30° . The two triangles have the same angles, so they are similar. The answer is yes.

### Angles, perimeters and lengths in similar triangles

- Example: The triangles are similar: ΔABC ∼ ΔDEF . In the first triangle, ∢A = 30°, ∢B = 115° . What size is ∢D ?
- Answer: 30 °
- Solution: ∢D matches ∢A , so it is 30°.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 9 topics](https://kixplo.com/uk/en/maths/year-9/)