# Similar triangles · Grade 10 · 4 units

> Similar triangles, Grade 10 · 4 units: The similarity tests and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Similar triangles (Grade 10 · 4 units) you practise 5 skills: The similarity tests · Corresponding sides · The scale factor and special lines · Similarity in a figure · From the ratio of areas to the ratio of sides.

Two triangles are similar when their angles are equal in pairs and their corresponding sides are in proportion. To show similarity, one of three tests is enough: two equal angles (AA), three sides in proportion (SSS), or two sides in proportion with the angle between them equal (SAS). An angle that is not between the two sides is not enough.

The ratio of corresponding sides is the scale factor. The perimeters are in the same ratio, and so are corresponding medians, heights and angle bisectors. The areas are in the ratio of the square of the scale factor: when the sides are in the ratio two to three, the areas are in the ratio four to nine. So from the ratio of the areas you find the scale factor by taking a square root.

A model proof: a line parallel to one side of a triangle cuts the other two sides. The angle at the vertex is shared by both triangles, and the angles next to the parallel line are equal as corresponding angles between parallel lines. So by AA the small triangle is similar to the large one.

**Tip:** Mark equal angles in the same colour: then you can see at once which sides correspond.

## What you learn

### The similarity tests

- Example: (Two triangles, one larger than the other.) In triangles ΔABC, ΔDEF we know that ∢B = ∢E = 60°, ∢A = ∢D = 75° . Which rule shows that the triangles are similar?
- Options: angle, angle AA · side, side, side SSS · side, angle, side SAS · Not enough information
- Answer: angle, angle
- Solution: Two angles of one triangle are equal to two angles of the other, so the triangles are similar by AA.

### Corresponding sides

- Example: (Two similar triangles, one larger than the other.) Triangles ΔABC ∼ ΔDEF are similar. The ratio of their sides is AB : DE = 2 : 3 . In centimetres, DE = 12 . How long is AB ?
- Answer: 8 cm
- Solution: Sides DE, AB match, so AB = 12 × 2/3 = 8 . It is 8 cm long.

### The scale factor and special lines

- Example: Triangle DEF is similar to triangle ABC , with the scale factor DE : AB equal to 5/2 . The perimeter of ABC is 16 cm. What is the perimeter of DEF ?
- Answer: 40 cm
- Solution: The perimeters are in the ratio of the scale factor: 16 × 5/2 = 40 . It is 40 cm.

### Similarity in a figure

- Example: (A triangle with a line inside it parallel to the base. The given lengths are written on the sides.) In the picture, DE ∥ BC . In centimetres, AD = 2, DB = 6, BC = 24 . What is the length of DE ?
- Answer: 6 cm
- Solution: The triangles are similar: ΔADE ∼ ΔABC . Side AB = 2 + 6 = 8 , and the scale factor: AD : AB = 2 : 8 = 1 : 4 . So DE = 24 ÷ 4 = 6 . The length is 6 cm.

### From the ratio of areas to the ratio of sides

- Example: Triangles ΔABC ∼ ΔDEF are similar. The area of ABC is 9 cm², and the area of DEF is 36 cm². Side DE = 12 cm. How long is AB ?
- Answer: 6 cm
- Solution: The ratio of the areas: 9 : 36 = 1 : 4 . The scale factor is its square root: AB : DE = √1 : √4 = 1 : 2 . So AB = 12 ÷ 2 = 6 . It is 6 cm.

## Not on the site yet

- Proving the similarity tests
- Proofs that use similarity
- Similar polygons in general

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 4 units topics](https://kixplo.com/il/en/math/grade-10-4u/)