Pythagorean Triads
A Pythagorean triad is a set of three whole numbers, such as \((3, 4, 5)\) or \((8, 15, 17)\), that fit \(a^2 + b^2 = c^2\). This NSW Year 9 Core maths subtopic revises triads from Stage 4 (NESA outcome MA4‑PYT‑C‑01) and looks at how new triads are made from old ones.
You will test sets such as \((13, 60, 61)\) and give a reason, recognise the common triads, find a missing member such as \(k\) in \((20, k, 29)\), and explain why \((1.5, 2, 2.5)\) makes a right-angled triangle but is not a triad. You will also see that multiplying every number by the same whole number gives another triad, while adding the same number does not. Triads give whole-number measurements for setting out square corners. It is part of Year 9 Core Pythagoras Theorem.
Every question with a fully worked solution.
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Theory
A Pythagorean triad is a set of three whole numbers that fit Pythagoras' theorem, so they can be the sides of a right-angled triangle. This Year 9 maths page shows how to test a set, find a missing member, and make new triads by multiplying one you already know.
A Pythagorean triad is three whole numbers \(a\), \(b\) and \(c\), with \(c\) the largest, such that
To test a set, use the same test as for a right-angled triangle. Square the two smaller numbers and add. Square the largest. If the results are equal, and all three are whole numbers, the set is a triad.
Every triad gives a right-angled triangle, with the largest number as the hypotenuse. But not every right-angled triangle gives a triad. A triangle with sides \(0.9\), \(1.2\) and \(1.5\) is right-angled, but those are not whole numbers, so they are not a triad.
Multiplying every member of a triad by the same whole number gives another triad. Adding the same number to every member does not.
Tradespeople use triads to set out square corners with just a tape measure.
Three whole numbers \(a\), \(b\), \(c\), with \(c\) the largest, form a triad when:
Multiply every member by the same whole number \(k\) and every square is multiplied by \(k^2\), so the equation still balances:
Four triads, each passing the test, and what each becomes when doubled:
| Triad | The test | Doubled |
|---|---|---|
| \((28,\ 45,\ 53)\) | \(784 + 2025 = 2809\) | \((56,\ 90,\ 106)\) |
| \((33,\ 56,\ 65)\) | \(1089 + 3136 = 4225\) | \((66,\ 112,\ 130)\) |
| \((48,\ 55,\ 73)\) | \(2304 + 3025 = 5329\) | \((96,\ 110,\ 146)\) |
| \((36,\ 77,\ 85)\) | \(1296 + 5929 = 7225\) | \((72,\ 154,\ 170)\) |
How to test or complete a Pythagorean triad
- Check that all three numbers are whole numbers. If not, it is not a triad.
- Order the numbers and pick out the largest.
- Square the two smaller numbers and add. Square the largest.
- Compare. Equal: a triad. Not equal: not a triad.
- For a missing member, rearrange the rule (add for the largest, subtract for a smaller one) and take the square root.
All three are whole numbers, so the test can be applied. The largest is \(89\). Square the two smaller numbers and add, then square the largest:
| \(39^2 + 80^2\) | \(=\) | \(1521 + 6400\) |
| \(\) | \(=\) | \(7921\) |
| \(89^2\) | \(=\) | \(7921\) |
They are equal, so yes, \((39, 80, 89)\) is a Pythagorean triad. A triangle with these sides is right-angled.
The largest is \(13\):
| \(8^2 + 9^2\) | \(=\) | \(64 + 81\) |
| \(\) | \(=\) | \(145\) |
| \(13^2\) | \(=\) | \(169\) |
\(145 \ne 169\), so no, it is not a triad. A triangle with sides \(8\), \(9\) and \(13\) is not right-angled.
\(39\) is the largest number, so it is the hypotenuse, and \(k\) is one of the two smaller numbers. Subtract:
| \(k^2\) | \(=\) | \(39^2 - 15^2\) |
| \(\) | \(=\) | \(1521 - 225\) |
| \(\) | \(=\) | \(1296\) |
| \(k\) | \(=\) | \(\sqrt{1296}\) |
| \(\) | \(=\) | \(\mathbf{36}\) |
\(36\) is a whole number, so \((15, 36, 39)\) really is a triad. If the largest member were the missing one, you would add the squares instead.
Multiply each member: \((33,\ 56,\ 65) \to (66,\ 112,\ 130)\). Then test the new set:
| \(66^2 + 112^2\) | \(=\) | \(4356 + 12\,544\) |
| \(\) | \(=\) | \(16\,900\) |
| \(130^2\) | \(=\) | \(16\,900\) |
Yes. Doubling every side makes every square \(2^2 = 4\) times bigger, so the two sides of the test stay equal.
Common pitfalls
Frequently asked questions
What is a Pythagorean triad?
Three whole numbers where the squares of the two smaller ones add up to the square of the largest. They can be the sides of a right-angled triangle. An example is \((39, 80, 89)\).
How do you check if three numbers are a Pythagorean triad?
Make sure they are whole numbers. Square the two smaller ones and add. If the total equals the square of the largest, they are a triad.
Are multiples of a Pythagorean triad also triads?
Yes. Multiplying every member by the same whole number multiplies every square by the same amount, so the equation still balances. \((28, 45, 53)\) doubled gives \((56, 90, 106)\), another triad.
Can a Pythagorean triad have decimals?
No. A triad must be three whole numbers. Sides such as \(0.9\), \(1.2\) and \(1.5\) make a right-angled triangle, but they are not a triad.
How do you find the missing number in a Pythagorean triad?
If the largest is missing, add the squares of the other two and take the square root. If a smaller one is missing, subtract the square of the other smaller one from the square of the largest, then take the square root.