Similar triangles · Year 11
In Similar triangles (Year 11) you practise 4 skills: The similarity tests · Corresponding sides · The scale factor and special lines · 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
For example: In triangles ΔABC, ΔDEF we know that ∢B = ∢E = 60°, ∢A = ∢D = 75° . Which rule shows that the triangles are similar?
- AA
- SSS
- SAS
- Not enough information
Answer and solution
The answer: angle, angle
Two angles of one triangle are equal to two angles of the other, so the triangles are similar by AA.
Corresponding sides
For example: Triangles ΔABC ∼ ΔDEF are similar. The ratio of their sides is AB : DE = 2 : 3 . In centimetres, DE = 12 . How long is AB ?
Answer and solution
The answer: 8 cm
Sides DE, AB match, so AB = 12 × 2/3 = 8 . It is 8 cm long.
The scale factor and special lines
For 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 and solution
The answer: 40 cm
The perimeters are in the ratio of the scale factor: 16 × 5/2 = 40 . It is 40 cm.
From the ratio of areas to the ratio of sides
For 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 and solution
The answer: 6 cm
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.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).