Similar triangles · Ages 14–15
Similar polygons
All angles equal, all sides in proportion; angles aren't multiplied.
How to do it
Method
All angles equal, all sides in proportion; angles aren't multiplied
k = 9 ÷ 6 = 1.5, QR = 4 × 1.5 = 6, ∠Q = ∠B
Another way
Write a proportion
QR4 = 96, 6 × QR = 36, QR = 6
In full
Two polygons are similar only when both hold: all corresponding angles equal, and all corresponding sides in proportion. One alone isn't enough: a square and a rectangle have all angles 90° but their sides aren't in proportion; a square and a rhombus have all sides equal but their angles needn't be. Triangles are the exception: equal angles force proportional sides. For ABCD ~ PQRS the letters match in order too: ∠A = ∠P, AB with PQ. One way: find the scale k = PQ ÷ AB and multiply every side by k; angles stay as they are. Another: write a proportion. Perimeters change by the same scale: a perimeter of 20 cm with k = 1.5 becomes 30 cm. Careful: all squares are similar, and all circles; but two rectangles needn't be.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A rectangle 7 cm long and 5 cm wide is similar to one 14 cm long and 11 cm wide.
True or false?
Show the answer and working
Answer: False
Rectangles' angles are all 90°, so look at the sides: similar only if 147 and 115 are equal.
Example 2
In the picture, KLMN ~ PQRS. x = , y =
Fill in each blank
Show the answer and working
Answer: 4, 100
Scale = 3 ÷ 6 = 0.5: x = 8 × 0.5 = 4; y is ∠Q, equal to ∠L = 100° — angles aren't scaled.
Example 3
A photo 15 cm long and 5 cm wide is enlarged into a similar one 8 cm wide. Its length =
Show the answer and working
Answer: 24 cm
Width with width, length with length: the scale is 8 ÷ 5, so the length = 15 × 8 ÷ 5 = 24 cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A rectangle 12 cm long and 3 cm wide is similar to one 18 cm long and 4.5 cm wide.
True or false?
Answer:
2.Which pair of figures must be similar?
- A. two squares
- B. two rectangles
- C. two rhombuses
- D. two isosceles triangles
Choose the right answer
Answer:
3.In the picture, ABCD ~ PQRS. x =
Answer:
4.A photo 10 cm long and 7 cm wide is enlarged into a similar one 14 cm wide. Its length =
Answer:
5.In the picture, KLMN ~ PQRS. x = , y =
Fill in each blank
Answer:
All the answers
- 1. True
- 2. A. two squares
- 3. 65°
- 4. 20 cm
- 5. 9, 105
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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