Similar triangles · Ages 14–15
Corresponding sides of similar triangles
Match the letters in order; big side = small side × the scale.
How to do it
Method
Match the letters in order; big side = small side × the scale
△ABC ~ △PQR, PQ ÷ AB = 12 ÷ 4 = 3, QR = 5 × 3 = 15
Another way
Write a proportion
x5 = 124, 4x = 60, x = 15
In full
Two triangles with the same shape, though maybe not the same size, are similar (written ~). Their corresponding angles are equal and their corresponding sides are in proportion. In △ABC ~ △PQR the vertices are written in matching order: A goes with P, B with Q, C with R, so AB goes with PQ, BC with QR, CA with RP. One way: find the scale first. AB = 4 cm and PQ = 12 cm: PQ ÷ AB = 3, so every side of the big triangle is 3 times its partner, and BC = 5 cm gives QR = 5 × 3 = 15 cm. Another: write a proportion. QRBC = PQAB, x5 = 124, cross-multiply: 4x = 60, x = 15. The two agree. The scale needn't be whole: x7 = 106 gives x = 706 = 353. Careful: use corresponding sides, not sides that happen to sit in the same place; a triangle may be turned or flipped in the picture, so use the equal angles to match the vertices.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
△ABC ~ △DEF, AB = 4 cm, BC = 5 cm and DE = 16 cm. EF =
Show the answer and working
Answer: 20 cm
DE goes with AB: the scale is 16 ÷ 4 = 4; EF goes with BC: EF = 5 × 4 = 20 cm.
Example 2
△ABC ~ △PRQ, with AB = 5 cm, BC = 9 cm, CA = 8 cm and PR = 15 cm. How long is QR?
- A. 19 cm
- B. 18 cm
- C. 24 cm
- D. 27 cm
Choose the right answer
Show the answer and working
Answer: D. 27 cm
A goes with P, B with R, C with Q. The scale is PR ÷ AB = 15 ÷ 5 = 3, and QR goes with BC: 9 × 3 = 27 cm.
Example 3
△ABC ~ △PQR with AB : PQ = 2 : 3. The perimeter of △ABC is 60 cm. The perimeter of △PQR =
Show the answer and working
Answer: 90 cm
Every side is multiplied by 32, so their sum is too: 60 × 3 ÷ 2 = 90 cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.△ABC ~ △PRQ, with AB = 8 cm, BC = 5 cm, CA = 6 cm and PR = 16 cm. How long is QR?
- A. 18 cm
- B. 13 cm
- C. 12 cm
- D. 10 cm
Choose the right answer
Answer:
2.In the picture, △ABC ~ △XYZ. x = , y =
Fill in each blank
Answer:
3.The two triangles in the picture are similar; equal angles carry the same marks. x =
Answer:
4.△ABC ~ △PQR, with AB = 9 cm, BC = 12 cm and PQ = 13 cm. QR = cm (as a fraction)
Answer in simplest form
Answer:
5.△ABC ~ △PQR with AB : PQ = 3 : 7. The perimeter of △ABC is 45 cm. The perimeter of △PQR =
Answer:
All the answers
- 1. D. 10 cm
- 2. 4.5, 4.5
- 3. 12 cm
- 4. 523
- 5. 105 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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