Pythagorean Triple
In short: A Pythagorean triple is a trio of counting numbers (a, b, c) with a² + b² = c², which means a right triangle can have exactly those three side lengths.
8th GradeGeometry8.G.B.7
What is Pythagorean Triple?
Most right triangles have a messy hypotenuse such as \(\sqrt{58}\), but a few have all three sides as whole numbers, and each such set of side lengths is a Pythagorean triple.
The numbers are written smallest to largest, like \((5, 12, 13)\), with the last one as the hypotenuse. Multiplying every number of a triple by the same whole number gives another triple: doubling \((5, 12, 13)\) yields \((10, 24, 26)\).
How to spot it
- A right triangle problem has side lengths that are all whole numbers with no decimals.
- You are asked to check whether three given lengths make a right triangle.
- A carpenter or builder wants a square corner using only a tape measure and a known set of lengths.
- A familiar trio of lengths, or a scaled-up copy of one, shows up in the diagram.
The method
- Order the three numbers from least to greatest and call them a, b and c.
- Compute \(a^2 + b^2\) and, separately, \(c^2\).
- If both results match, the numbers form a triple; if not, they do not.
- To create more, multiply all three numbers of a known triple by one whole number.
Pythagorean Triple: solved example
Problem
A fence builder in Wichita wants to check a corner using stakes 12 ft and 35 ft along the two fences, with a 37 ft cord between the stake tops. Is the corner square?
Solution
Order the lengths: 12, 35, 37. Then \(12^2 + 35^2 = 144 + 1{,}225 = 1{,}369\), and \(37^2 = 1{,}369\).
The values match, so \((12, 35, 37)\) is a Pythagorean triple and the corner is square, a true \(90^{\circ}\) angle.
Try it
Question 1
Multiply the triple \((5, 12, 13)\) by 6 and verify that the result still works.
Show the answer
\((30, 72, 78)\): \(900 + 5{,}184 = 6{,}084 = 78^2\). It works.
Question 2
Do 6, 8 and 11 form a triple?
Show the answer
\(36 + 64 = 100\), but \(11^2 = 121\). They do not match, so no.
Pythagorean Triple vs. Pythagorean Theorem
The theorem is the rule \(a^2 + b^2 = c^2\) that every right triangle obeys, whatever its sides look like. A triple is a special set of whole numbers that happens to satisfy the rule exactly. All triples come from the theorem, but most right triangles are not built on a triple.
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