Similar triangles in context · Year 10
In Similar triangles in context (Year 10) you practise 5 skills: The enlargement factor · Are the triangles similar · Sides of similar triangles · Perimeters and areas of similar triangles · Indirect measurement. Similar shapes are shapes where one is an enlargement or a reduction of the other: a photo and its print, a drawing and its copy, a big sail and a small one. Every length grows by the same number, called the enlargement factor, and the angles stay the same. A factor above one enlarges, and below one it reduces. Adding the same length to every side does not give a similar shape.
To check that two triangles are similar, it is enough to find two equal angles in both, since the third angle is then equal too. You can also sort the sides of each triangle from short to long and check that each pair of matching sides gives the same result when divided. When triangles are written as similar, the vertices match in the order written. Find a missing side from the factor: divide two known matching sides, then multiply by it. In a right triangle, the third side then comes from Pythagoras’ theorem. The perimeter grows by the factor, but the area grows by the factor squared: when the sides grow three times, the area grows nine times. Similarity helps measure lengths that are hard to reach. At the same hour, a pole and a tree cast shadows, and the triangles they make with their shadows are similar. A mirror on the ground, a camera and the width of a river also give two similar triangles: one with every length known, the other with the length asked for. Write every length in the same unit before working.
Tip: In similar triangles, every side divided by its matching side gives the same factor.
What you learn
The enlargement factor
For example: A photo 10 cm by 15 cm is enlarged to 40 cm by 60 cm. What is the enlargement factor?
Answer and solution
The answer: 4
Divide a new length by the matching original length: 40 ÷ 10 = 4 The other pair gives the same factor: 60 ÷ 15 = 4
Are the triangles similar
For example: In triangle ABC : ∢A = 115°, ∢B = 35° . In triangle DEF : ∢D = 115°, ∢E = 35° . Are the triangles similar?
- Yes
- No
Answer and solution
The answer: Yes
Work out the third angle in each triangle: ∢C = 180° − 115° − 35° = 30° ∢F = 180° − 115° − 35° = 30° . The two triangles have the same angles, so they are similar. The answer is yes.
Sides of similar triangles
For example: The triangles are similar: ΔABC ∼ ΔDEF . The lengths are in centimetres. In the first triangle, AB = 6, BC = 10, AC = 7 . In the second triangle, DF = 28 . What is the length of side EF ?
Answer and solution
The answer: 40 cm
Side DF matches side AC , so the scale factor is 28 ÷ 7 = 4 . The side you want matches side BC : EF = 10 × 4 = 40 . The length of the side is 40 cm.
Perimeters and areas of similar triangles
For example: Two kites are similar triangles. The perimeter of the big kite is 114 cm. The ratio of a side of the small kite to the matching side of the big one is 2 : 3 . What is the perimeter of the small kite?
Answer and solution
The answer: 76 cm
The small perimeter is the big one times the ratio: 114 × 2 ÷ 3 = 76 The perimeter is 76 cm.
Indirect measurement
For example: At the same hour, an upright pole 2 m tall casts a shadow 6 m long, and a tree casts a shadow 24 m long. How tall is the tree?
Answer and solution
The answer: 8 m
The tree’s shadow is longer by a factor of: 24 ÷ 6 = 4 The tree is taller than the pole by the same factor: 2 × 4 = 8 The tree is 8 m tall.
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).