Pythagoras' theorem · Ages 13–14
The converse of Pythagoras' theorem
Short sides' squares add to the longest's square ⇒ a right angle opposite it.
How to do it
Method
Short sides' squares add to the longest's square ⇒ a right angle opposite it
92 + 122 = 225 = 152
Another way
Simplify first
9, 12, 15 ÷ 3 = 3, 4, 5
In full
The converse of Pythagoras' theorem: if a triangle's sides a, b and c (c the longest) fit a2 + b2 = c2, it is right-angled, and the right angle faces the longest side c. To decide, find the longest side, then compare the squares of the two shorter sides added with the square of the longest: equal, a right-angled triangle; not equal, not one. 9, 12, 15: 92 + 122 = 81 + 144 = 225 = 152, so it is right-angled, the right angle facing the side 15 long. Keep the reasons apart: from a right angle to the sides is Pythagoras' theorem; from the three sides to a right angle is its converse. Or simplify first: 9, 12 and 15 divided by 3 are 3, 4, 5, a Pythagorean triple, so the triangle is right-angled.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A triangle's sides are 33 cm, 30 cm and 16 cm. The longest is cm; the two shorter sides' squares add to , and the longest side's square is .
Fill in each blank
Show the answer and working
Answer: 33, 1156, 1089
The longest is 33 cm; the other two squared and added make 1156, and 332 = 1089. Only when the two are equal is the triangle right-angled.
Example 2
A triangle with sides of 72 cm, 30 cm and 78 cm is right-angled. How long is the side opposite the right angle, in cm?
Show the answer and working
Answer: 78 cm
The right angle faces the longest side: 722 + 302 = 6084 = 782, so it is 78 cm.
Example 3
Which of these sets of lengths (in cm) makes a right-angled triangle?
- A. 64, 60, 87
- B. 63, 60, 87
- C. 63, 60, 88
- D. 63, 61, 88
Choose the right answer
Show the answer and working
Answer: B. 63, 60, 87
632 + 602 = 7569 = 872, right-angled by the converse of Pythagoras' theorem; the other three don't match.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A triangle with sides of 4 cm, 5 cm and 3 cm is right-angled. How long is the side opposite the right angle, in cm?
Answer:
2.A triangle's sides are 6 cm, 10 cm and 8 cm. The longest is cm; the two shorter sides' squares add to , and the longest side's square is .
Fill in each blank
Answer:
3.In triangle ABC, AB = 30 cm, BC = 34 cm and CA = 16 cm. Which angle is the right angle?
- A. ∠B
- B. ∠A
- C. None of them
- D. ∠C
Choose the right answer
Answer:
4.In triangle PQR, PQ = 48 cm, QR = 90 cm and PR = 102 cm. Prove ∠PQR = 90°: put the steps in order.
- A. = 2304 + 8100
- B. = 10404 = 1022 = PR2
- C. PQ2 + QR2 = 482 + 902
- D. ∠PQR = 90° (converse of Pythagoras' theorem)
Write the letters in the right order
Answer:
5.A triangle with sides of 65 cm, 63 cm and 16 cm is right-angled.
True or false?
Answer:
All the answers
- 1. 5 cm
- 2. 10, 100, 100
- 3. B. ∠A
- 4. C → A → B → D
- 5. True
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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