For the odd number n = 3: (n^2 − 1) ÷ 2 = □ and (n^2 + 1) ÷ 2 = □. The three make a Pythagorean triple. Fill in each blank

4, 5. 3^2 = 9, (9 − 1) ÷ 2 = 4 and (9 + 1) ÷ 2 = 5; they are 1 apart, so 5^2 − 4^2 = 5 + 4 = 9 = 3^2.

Which of these is a Pythagorean triple? A. 21, 22, 30 B. 20, 21, 30 C. 20, 21, 29 D. 20, 22, 30 Choose the right answer

C. 20, 21, 29. 20^2 + 21^2 = 841 = 29^2. Multiplying by one number keeps a triple; adding one doesn't.

In a Pythagorean triple the smallest number is the odd number 17, and the other two are 1 apart. What is the largest?

145. Call the other two y and y + 1: (y + 1)^2 − y^2 = 2y + 1 = 17^2 = 289, so y = 144 and the largest is 145.

Pythagoras' theorem · Ages 13–14

Pythagorean triples

m² − n², 2mn, m² + n²; or odd n, (n² − 1) ÷ 2, (n² + 1) ÷ 2.

How to do it

Method

m^2 − n^2, 2mn, m^2 + n^2; or odd n, (n^2 − 1) ÷ 2, (n^2 + 1) ÷ 2

m = 3, n = 2: 5, 12, 13

Another way

The odd-number rule

n = 7: 7, 24, 25

In full

Three whole numbers a, b and c with a² + b² = c² are a Pythagorean triple: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25. Multiples of a triple are triples too: 3, 4, 5 times k gives 3k, 4k, 5k. To make new ones, take two whole numbers m > n: a = m² − n², b = 2mn, c = m² + n². With m = 3 and n = 2: a = 9 − 4 = 5, b = 12, c = 9 + 4 = 13; check: 5² + 12² = 169 = 13². Another way: for any odd number n above 1, n, (n² − 1) ÷ 2 and (n² + 1) ÷ 2 are a triple, the last two 1 apart; n = 7: (49 − 1) ÷ 2 = 24 and (49 + 1) ÷ 2 = 25, giving 7, 24, 25.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
For the odd number n = 3: (n² − 1) ÷ 2 = and (n² + 1) ÷ 2 = . The three make a Pythagorean triple.
Fill in each blank
Show the answer and working
Answer: 4, 5
3² = 9, (9 − 1) ÷ 2 = 4 and (9 + 1) ÷ 2 = 5; they are 1 apart, so 5² − 4² = 5 + 4 = 9 = 3².
Example 2
Which of these is a Pythagorean triple?
- A. 21, 22, 30
- B. 20, 21, 30
- C. 20, 21, 29
- D. 20, 22, 30
Choose the right answer
Show the answer and working
Answer: C. 20, 21, 29
20² + 21² = 841 = 29². Multiplying by one number keeps a triple; adding one doesn't.
Example 3
In a Pythagorean triple the smallest number is the odd number 17, and the other two are 1 apart. What is the largest?
Show the answer and working
Answer: 145
Call the other two y and y + 1: (y + 1)² − y² = 2y + 1 = 17² = 289, so y = 144 and the largest is 145.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.24, 45, 52 is a Pythagorean triple.
True or false?
Answer:
2.For the odd number n = 5: (n² − 1) ÷ 2 = and (n² + 1) ÷ 2 = . The three make a Pythagorean triple.
Fill in each blank
Answer:
3.Make a Pythagorean triple with m = 4 and n = 3: m² − n² = , 2mn = , m² + n² = .
Fill in each blank
Answer:
4.32, 60 and one more number make a Pythagorean triple, the last the biggest of the three. What is it?
Answer:
5.In a Pythagorean triple the smallest number is odd and the other two are 1 apart, the largest being 85. What is the smallest?
Answer:

All the answers

1. False
2. 12, 13
3. 7, 24, 25
4. 68
5. 13

Free worksheet: Pythagorean triplesWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

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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