Testing for Pythagorean Triads
A Pythagorean triad is a set of three whole numbers, such as \((5, 12, 13)\), where the squares of the two smaller numbers add up to the square of the largest. In NSW Year 8 maths, triads give right-angled triangles whose sides are all whole numbers (NESA outcome MA4‑PYT‑C‑01).
You will test sets of numbers, including sets not written in order, and give a reason for each answer. You will find a missing number in a triad, build new triads by multiplying \((3, 4, 5)\) or \((8, 15, 17)\) by a whole number, and see why adding two triads does not work. Tilers and landscapers use triads to set out square corners with no rounding. It is part of Year 8 Pythagoras.
Theory
A Pythagorean triad is three whole numbers \(a\), \(b\), \(c\) with \(a^2 + b^2 = c^2\), where \(c\) is the largest. A triad gives a right-angled triangle whose sides are all whole numbers. This Year 8 maths page shows how to test a set, find a missing number and build new triads.
Three whole numbers \(a\), \(b\) and \(c\) are a Pythagorean triad when \(a^2 + b^2 = c^2\), where \(c\) is the largest of the three. For example \((3, 4, 5)\): \(9 + 16 = 25\).
The test. Pick out the largest number: that is \(c\). Square the two smaller numbers and add. Square the largest number. Equal means yes; not equal means no.
Any order. The largest number is \(c\) wherever it is written: \((53, 28, 45)\) is tested as \(28^2 + 45^2\) against \(53^2\).
Whole numbers only. Decimals such as \(1.5\), \(2\), \(2.5\) fit \(a^2 + b^2 = c^2\) and make a right-angled triangle, but they are not a triad.
Multiples. Multiply all three numbers of a triad by the same whole number and you get another triad: doubling \((3, 4, 5)\) gives \((6, 8, 10)\). Adding two triads does not give a triad.
Missing number. Missing the largest: add the squares of the two smaller, then take the square root. Missing a smaller one: subtract from the square of the largest, then take the square root. A whole-number answer makes a triad.
A pattern. Start with an odd number \(n\). The middle number is \((n^2 - 1) \div 2\) and the largest is one more. Starting with \(3\) gives \((3, 4, 5)\); starting with \(9\) gives \((9, 40, 41)\).
Common Pythagorean triads (and their multiples):
| Triad | Check | A multiple |
|---|---|---|
| \((3, 4, 5)\) | \(9 + 16 = 25\) | \((6, 8, 10)\) |
| \((5, 12, 13)\) | \(25 + 144 = 169\) | \((10, 24, 26)\) |
| \((8, 15, 17)\) | \(64 + 225 = 289\) | \((16, 30, 34)\) |
| \((7, 24, 25)\) | \(49 + 576 = 625\) | \((14, 48, 50)\) |
| \((9, 40, 41)\) | \(81 + 1600 = 1681\) | \((18, 80, 82)\) |
| \((20, 21, 29)\) | \(400 + 441 = 841\) | \((40, 42, 58)\) |
How to test for a Pythagorean triad
- Check all three are whole numbers.
- Pick the largest as \(c\).
- Work out \(a^2 + b^2\) and \(c^2\).
- Compare: equal means it is a triad.
How to find a missing number
- Largest missing: \(c = \sqrt{a^2 + b^2}\).
- Smaller missing: \(a = \sqrt{c^2 - b^2}\).
- Check the answer is a whole number.
| \(17^2 + 144^2\) | \(=\) | \(289 + 20\,736\) |
| \(\) | \(=\) | \(21\,025\) |
| \(145^2\) | \(=\) | \(21\,025\) |
Equal, so \((17, 144, 145)\) is a Pythagorean triad.
| \(11^2 + 110^2\) | \(=\) | \(121 + 12\,100\) |
| \(\) | \(=\) | \(12\,221\) |
| \(111^2\) | \(=\) | \(12\,321\) |
\(12\,221 \ne 12\,321\), so \((11, 110, 111)\) is not a triad.
The largest number is \(20\), so \(20\) is \(c\), even though it is written first.
| \(12^2 + 15^2\) | \(=\) | \(144 + 225 = 369\) |
| \(20^2\) | \(=\) | \(400\) |
\(369 \ne 400\), so it is not a triad.
(i) Multiply each number by \(3\).
(ii) Show that the new set is a triad.
| \(12^2 + 35^2\) | \(=\) | \(144 + 1225\) |
| \(\) | \(=\) | \(1369 = 37^2\) |
| \(36^2 + 105^2\) | \(=\) | \(1296 + 11\,025\) |
| \(\) | \(=\) | \(12\,321\) |
| \(111^2\) | \(=\) | \(12\,321\) |
(i) \((36, 105, 111)\)
(ii) Both sides give \(12\,321\), so it is a triad.
\(73\) is the largest, so the missing number is a smaller one: subtract.
| \(x^2\) | \(=\) | \(73^2 - 48^2\) |
| \(\) | \(=\) | \(5329 - 2304 = 3025\) |
| \(x\) | \(=\) | \(\sqrt{3025} = 55\) |
\(x = 55\). Check: \(48^2 + 55^2 = 2304 + 3025 = 5329 = 73^2\).
(ii) How many triads \((20k, 21k, 29k)\), with \(k\) a whole number from \(1\), have a largest number less than \(100\)?
| \((13^2 - 1) \div 2\) | \(=\) | \(168 \div 2\) |
| \(\) | \(=\) | \(84\) |
| \(13^2 + 84^2\) | \(=\) | \(169 + 7056\) |
| \(\) | \(=\) | \(7225 = 85^2\) |
(i) \((13, 84, 85)\)
(ii) \(29 \times 3 = 87 < 100\) but \(29 \times 4 = 116\), so \(k = 1, 2, 3\): \(3\) triads.
Common pitfalls
Frequently asked questions
What is a Pythagorean triad?
Three whole numbers \(a\), \(b\), \(c\) with \(a^2 + b^2 = c^2\), such as \((3, 4, 5)\).
How do you test for a Pythagorean triad?
Square the two smaller numbers and add. If the total equals the square of the largest, it is a triad.
Is a multiple of a triad also a triad?
Yes. Multiplying \((12, 35, 37)\) by \(2\) gives \((24, 70, 74)\), and \(576 + 4900 = 5476 = 74^2\).
How do you find a missing number in a triad?
Add the squares and take the root for the largest; subtract and take the root for a smaller one.
Why are triads useful?
They give right-angled triangles with whole-number sides, so builders can set out a square corner with no rounding.