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Over 2,000 years ago, the Greek philosopher Pythagoras noticed something amazing about right-angled triangles:
A 3–4–5 triangle. 9 + 16 = 25 ✓
A 5–12–13 triangle. 25 + 144 = 169 ✓
The rule is: a² + b² = c², where a and b are the two short sides and c is the longest — the hypotenuse.
In the next sections you'll investigate which sets of whole numbers actually make this rule work — we call those Pythagorean triples.
A Pythagorean triple is a set of three positive whole numbers (a, b, c) that make a² + b² = c² true. The biggest number is always c — the hypotenuse.
3² + 4² = 9 + 16 = 25
5² = 25
Equal → (3, 4, 5) IS a triple ✓
6² + 8² = 36 + 64 = 100
11² = 121
Not equal → (6, 8, 11) is NOT a triple ✗
Here's the big discovery: if you multiply every side of a Pythagorean triple by the same whole number, you get another Pythagorean triple.
(3, 4, 5) × 2 = (6, 8, 10)
Check: 6² + 8² = 36 + 64 = 100 and 10² = 100. Equal ✓
(3, 4, 5) × 3 = (9, 12, 15) → still a triple
If we multiply (3, 4, 5) by 1.2, we get (3.6, 4.8, 6). The right-triangle rule still works: 3.6² + 4.8² = 36 = 6². But it's no longer a triple — a Pythagorean triple must be three whole numbers.
Going backwards: if you're given a scaled-up triple and one of its sides is missing, you can find it by working out the scale factor from a known triple.
The other direction: given a big triple like (210, 280, 350), find the simpler triple it came from by dividing by a common factor.
210 ÷ 70 = 3, 280 ÷ 70 = 4, 350 ÷ 70 = 5. So (210, 280, 350) = (3, 4, 5) × 70.
The hypotenuse is the longest side of a right-angled triangle, and it is always the side opposite the right angle — the side that does not touch the 90° corner.
Notice: the hypotenuse doesn't care which way the triangle is turned. It is always opposite the square marker.
Tap the side that is opposite the right angle.
If you know the two short sides (legs), you can find the hypotenuse. Square both short sides, add them, then take the square root.
If you know the hypotenuse and one short side, you can find the other short side. Same rule, rearranged: subtract the known short side's square from the hypotenuse's square, then take the square root.
You have worked through all three sections of Pythagoras' Theorem.