Key takeaway: A Pythagorean triple is a set of three positive whole numbers a, b and c that satisfy a squared plus b squared equals c squared, so they are the sides of a right triangle.
Grade 88.G.B.7Geometry and Measurement
Definition of Pythagorean Triple
A Pythagorean triple is a group of three positive integers that fit the Pythagorean equation exactly. Every such group describes a right triangle whose sides are whole numbers, with no decimals or square roots left over.
Multiplying all three numbers by the same whole number gives a new triple, so each basic triple starts a whole family. The set 8, 15, 17 yields 16, 30, 34 and 24, 45, 51. A triple that cannot be divided by a common factor is called primitive.
Formula and notation
Write a triple in order, smallest to largest, with the hypotenuse last:
\[(a, b, c) \quad\text{with}\quad a^{2} + b^{2} = c^{2}\]
For example \((8, 15, 17)\) works because \(64 + 225 = 289 = 17^{2}\). If \((a, b, c)\) is a triple and \(k\) is a positive integer, then \((ka, kb, kc)\) is a triple.
Key facts
- Primitive triples include 3-4-5, 5-12-13, 8-15-17, 7-24-25 and 20-21-29.
- Scaling a triple by any whole number produces another triple.
- The numbers must be integers; 1.5, 2, 2.5 is a right triangle but not a triple.
- The largest number in a triple is always the hypotenuse.
Pythagorean Triple example
Decide whether 28, 45 and 53 form a Pythagorean triple.
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The largest number, 53, is the candidate hypotenuse.
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Square the two shorter numbers: \(28^{2} + 45^{2} = 784 + 2025 = 2809\).
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Square the largest: \(53^{2} = 2809\). The two results match.
Yes, 28, 45 and 53 are a Pythagorean triple.
Common mistake
Watch out
Beth tests 28, 45 and 53 by computing \(28^{2} + 53^{2} = 3593\) and comparing it with \(45^{2} = 2025\), then concludes it is not a triple.
Why it is wrong
She put the longest number on the wrong side of the equation. The hypotenuse is always the largest number, so only \(c^{2}\) can stand alone. Testing \(28^{2} + 45^{2}\) against \(53^{2}\) shows the equality \(2809 = 2809\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement