Triangles • Topic 4 of 5

Pythagorean Theorem

In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the two legs: a² + b² = c². Given the two legs, the hypotenuse is √(a² + b²); given the hypotenuse and one leg, the other leg is √(c² − a²). Memorising common Pythagorean triples — 3-4-5, 5-12-13, 8-15-17, 7-24-25 — and their multiples lets you solve many problems instantly without a calculator. The theorem only applies to right triangles, and the hypotenuse is always the longest side. It underlies distance problems and much of coordinate geometry on the SAT.

A 3-4-5 right triangle illustrating a² + b² = c²Pythagorean Theorem: a² + b² = c²3453² + 4² = 9 + 16 = 25 = 5²

✅ Solved examples

1. A right triangle has legs 3 and 4. Find the hypotenuse.
√(9 + 16) = √25 = 5.
2. Legs 6 and 8. Find the hypotenuse.
√(36 + 64) = √100 = 10.
3. Hypotenuse 13, one leg 5. Find the other leg.
√(169 − 25) = √144 = 12.
4. Legs 8 and 15. Find the hypotenuse.
√(64 + 225) = √289 = 17.

✏️ Practice — try these, take hints as needed

1. A right triangle has legs 9 and 12. Find the hypotenuse.
√(a² + b²).
√(81 + 144) = √225.
—
15.
2. Legs 5 and 12. Find the hypotenuse.
√(25 + 144).
√169.
—
13.
3. Hypotenuse 10, one leg 6. Find the other leg.
√(100 − 36).
√64.
—
8.
4. Legs 7 and 24. Find the hypotenuse.
√(49 + 576).
√625.
—
25.
5. Hypotenuse 17, one leg 15. Find the other leg.
√(289 − 225).
√64.
—
8.

📝 Topic test — 8 questions

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