Pythagoras' theorem · Ages 13–14
Finding a shorter side
A leg is a subtraction: b2 = c2 − a2.
How to do it
Method
A leg is a subtraction: b^2 = c^2 − a^2
132 − 52 = 144, 144 = 12
Another way
Difference of squares
132 − 52 = (13 + 5)(13 − 5) = 18 × 8 = 144
In full
Given the hypotenuse and one leg, to find the other leg turn the theorem round: b2 = c2 − a2, the square of the hypotenuse take away the square of the leg you know, then the root. Hypotenuse 13 cm, leg 5 cm: 132 − 52 = 169 − 25 = 144, so the other leg = 144 = 12 cm. Be sure which side is the hypotenuse: the one facing the right angle, the longest. Careful: for a leg you subtract, never add; added, the answer would come out longer than the hypotenuse, which can't be. Or use the difference of two squares: 132 − 52 = (13 + 5)(13 − 5) = 18 × 8 = 144, quicker with big numbers.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The right-angled triangle's hypotenuse is 25 cm and one leg 7 cm. How long is the leg ?, in cm?
- A. 24
- B. 576
- C. 32
- D. 18
Choose the right answer
Show the answer and working
Answer: A. 24
252 − 72 = 625 − 49 = 576, so ? = 576 = 24 cm.
Example 2
A leg by the difference of two squares: hypotenuse 39 cm, leg 36 cm, 392 − 362 = (39 + 36)(39 − 36) = × = , so the other leg = cm.
Fill in each blank
Show the answer and working
Answer: 75, 3, 225, 15
(39 + 36)(39 − 36) = 75 × 3 = 225, the same as 1521 − 1296; 225 = 15.
Example 3
Two sides of a right-angled triangle are 7 cm and 10 cm, but it doesn't say whether either is the hypotenuse. If the 10 cm side is the hypotenuse, the third side = cm; if both are legs, the third side = cm.
Fill in each blank
Show the answer and working
Answer: 51, 149
With 10 cm the hypotenuse the third side is a leg, so subtract: 100 − 49 = 51; with both legs it is the hypotenuse, so add: 49 + 100 = 149. The 7 cm side is shorter than 10 cm, so it can't be the hypotenuse.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The right-angled triangle's hypotenuse is 10 cm and one leg 6 cm. How long is the leg ?, in cm?
- A. 4
- B. 64
- C. 8
- D. 16
Choose the right answer
Answer:
2.In the right-angled triangle, how long is the leg ?, in cm?
Answer:
3.A leg by the difference of two squares: hypotenuse 123 cm, leg 120 cm, 1232 − 1202 = (123 + 120)(123 − 120) = × = , so the other leg = cm.
Fill in each blank
Answer:
4.In a right-angled triangle the hypotenuse c is 27 cm longer than the leg x, and the other leg is 45 cm. 452 = c2 − x2 = (c + x)(c − x), so c + x = ÷ 27 = and x = cm.
Fill in each blank
Answer:
5.Two sides of a right-angled triangle are 11 cm and 14 cm, but it doesn't say whether either is the hypotenuse. If the 14 cm side is the hypotenuse, the third side = cm; if both are legs, the third side = cm.
Fill in each blank
Answer:
All the answers
- 1. C. 8
- 2. 16 cm
- 3. 243, 3, 729, 27
- 4. 2025, 75, 24
- 5. 75, 317
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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