Pythagorean Triads
A Pythagorean triad is a set of three whole numbers, such as \((5, 12, 13)\), that fit \(a^2 + b^2 = c^2\). In NSW Year 9 Advanced maths, triads give right-angled triangles with whole-number sides and some neat algebra, revising Stage 4 (NESA outcome MA4‑PYT‑C‑01).
You will test sets of numbers, find a missing member of a triad, multiply a triad to make a new one and explain why adding \(1\) to each number fails. You will use triads to find lengths quickly, check whether a corner is square, and make triads from rules such as \(a = m^2 - n^2\), \(b = 2mn\), \(c = m^2 + n^2\), including the triad whose largest number is \(53\). It is part of Year 9 Advanced Pythagoras Theorem.
Every question with a fully worked solution.
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Theory
A Pythagorean triad is three whole numbers \(a\), \(b\), \(c\) with \(a^2 + b^2 = c^2\), such as \((3,\ 4,\ 5)\). This Year 9 Advanced maths page revises the earlier idea and extends it: testing triads, finding a missing number, multiples of a triad, formulas that generate triads, and equations where a member is written with \(x\).
A Pythagorean triad (or Pythagorean triple) is a set of three whole numbers that can be the sides of a right-angled triangle: the squares of the two smaller numbers add to the square of the largest. \((3,\ 4,\ 5)\) is the smallest: \(9 + 16 = 25\).
Both conditions matter. \((1.2,\ 1.6,\ 2)\) makes a right-angled triangle but is not a triad, because \(1.2\) and \(1.6\) are not whole numbers. \((4,\ 5,\ 6)\) is three whole numbers but not a triad, because \(16 + 25 \ne 36\).
Multiplying every number in a triad by the same whole number gives another triad: \((3,\ 4,\ 5) \times 2 = (6,\ 8,\ 10)\). A triad that is not a multiple of a smaller one is called primitive.
Triads are useful because they give exact whole-number answers. Spotting one, such as a multiple of \((3,\ 4,\ 5)\), lets you write down a third side without a calculator.
\((a,\ b,\ c)\) is a Pythagorean triad when \(a\), \(b\), \(c\) are whole numbers and
Two ways to generate triads:
How to work with Pythagorean triads
- Check all three numbers are whole numbers. Decimals or surds can make a right-angled triangle, but not a triad.
- Test by squaring the two smaller numbers, adding, and comparing with the square of the largest. Close is not enough.
- Find a missing number by adding squares (if it is the largest) or subtracting from the largest square. The answer must be a whole number.
- Scale by multiplying every number by the same whole number. Adding the same number to each does not work.
- Use a known triad to find lengths and areas without a calculator: its two smaller numbers are the base and height of the right-angled triangle.
(i) \((48,\ 55,\ 73)\)
(ii) \((21,\ 72,\ 76)\)
Square the two smaller numbers, add, and compare with the square of the largest.
(i)
| \(48^2 + 55^2\) | \(=\) | \(2304 + 3025 = 5329\) |
| \(73^2\) | \(=\) | \(5329\) |
Equal, and all three are whole numbers, so \((48,\ 55,\ 73)\) is a triad.
(ii)
| \(21^2 + 72^2\) | \(=\) | \(441 + 5184 = 5625\) |
| \(76^2\) | \(=\) | \(5776\) |
Not equal, so \((21,\ 72,\ 76)\) is not a triad. Since \(5625 = 75^2\), the set \((21,\ 72,\ 75)\) would be.
(i) \((39,\ k,\ 89)\) is a Pythagorean triad. Find \(k\).
(ii) \((12,\ 35,\ 37)\) is a triad. Find the triad with the same shape whose smallest number is \(60\).
(i) \(89\) is the largest, so it is the hypotenuse and \(k\) is found by subtracting.
| \(k^2\) | \(=\) | \(89^2 - 39^2\) |
| \(\) | \(=\) | \(7921 - 1521\) |
| \(\) | \(=\) | \(6400\) |
| \(k\) | \(=\) | \(\mathbf{80}\) |
(ii) The multiplier is \(60 \div 12 = 5\). Multiply every number by \(5\).
| \((12,\ 35,\ 37) \times 5\) | \(=\) | \(\mathbf{(60,\ 175,\ 185)}\) |
Check: \(60^2 + 175^2 = 3600 + 30\,625 = 34\,225 = 185^2\).
(i) Use \(a = m^2 - n^2\), \(b = 2mn\), \(c = m^2 + n^2\) with \(m = 9\) and \(n = 4\).
(ii) Use the odd-number rule \(n,\ \dfrac{n^2 - 1}{2},\ \dfrac{n^2 + 1}{2}\) with \(n = 17\). Check each triad.
(i) Substitute, then check.
| \(a\) | \(=\) | \(81 - 16 = 65\) |
| \(b\) | \(=\) | \(2 \times 9 \times 4 = 72\) |
| \(c\) | \(=\) | \(81 + 16 = 97\) |
| \(65^2 + 72^2\) | \(=\) | \(4225 + 5184 = 9409 = 97^2\) |
The triad is \(\mathbf{(65,\ 72,\ 97)}\).
(ii)
| \(\dfrac{17^2 - 1}{2}\) | \(=\) | \(\dfrac{288}{2} = 144\) |
| \(\dfrac{17^2 + 1}{2}\) | \(=\) | \(\dfrac{290}{2} = 145\) |
| \(17^2 + 144^2\) | \(=\) | \(289 + 20\,736 = 21\,025 = 145^2\) |
The triad is \(\mathbf{(17,\ 144,\ 145)}\). The two larger numbers differ by \(1\), which happens every time with this rule.
The largest number is the hypotenuse, so it goes on its own. Expand and the \(x^2\) terms cancel.
| \(x^2 + 28^2\) | \(=\) | \((x + 14)^2\) |
| \(x^2 + 784\) | \(=\) | \(x^2 + 28x + 196\) |
| \(588\) | \(=\) | \(28x\) |
| \(x\) | \(=\) | \(\mathbf{21}\) |
The triad is \((21,\ 28,\ 35)\). Check: \(441 + 784 = 1225 = 35^2\). It is \((3,\ 4,\ 5)\) multiplied by \(7\).
Common pitfalls
Frequently asked questions
What is a Pythagorean triad?
Three whole numbers \(a\), \(b\) and \(c\) where \(a^2 + b^2 = c^2\), so they can be the sides of a right-angled triangle. The smallest is \((3,\ 4,\ 5)\).
How do you check if three numbers are a Pythagorean triad?
Make sure all three are whole numbers. Then square the two smaller ones and add. If the total equals the square of the largest, the set is a triad.
Is a multiple of a Pythagorean triad also a triad?
Yes. If every number is multiplied by the same whole number \(k\), each square is multiplied by \(k^2\), so the equation still balances. For example, \((6,\ 8,\ 10)\) is \((3,\ 4,\ 5)\) doubled.
How do you make new Pythagorean triads?
Choose whole numbers \(m > n\) and use \(a = m^2 - n^2\), \(b = 2mn\), \(c = m^2 + n^2\). For an odd number \(n\), the set \(n,\ \dfrac{n^2 - 1}{2},\ \dfrac{n^2 + 1}{2}\) is also a triad.
Can a Pythagorean triad contain decimals?
No. A triad must be three whole numbers. Decimal sides can still make a right-angled triangle, but they are not a triad.