Triangles • Topic 3 of 6

The Pythagorean Theorem

In any 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. The GRE recycles a small set of whole-number right triangles — the Pythagorean triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25, plus their multiples (6-8-10, 9-12-15, 10-24-26, and so on). Spotting a triple on sight saves you from squaring and taking a root on the slow on-screen calculator. When no triple appears, apply the formula directly and leave irrational answers in exact radical form unless a decimal is requested. Always confirm which side is the hypotenuse: it is opposite the right angle and is the longest side.

A 6 by 8 by 10 right triangle with the right angle markedA 6-8-10 right triangle86106² + 8² = 10² · 36 + 64 = 100

✅ Solved examples

1. A right triangle has legs 6 and 8. Find the hypotenuse.
6² + 8² = 36 + 64 = 100, so the hypotenuse = √100 = 10 (a 3-4-5 triple doubled).
2. A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.
5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12 (the 5-12-13 triple).
3. A right triangle has legs 9 and 12. Find the hypotenuse.
This is 3-4-5 scaled by 3: 9 = 3×3, 12 = 3×4, so the hypotenuse = 3×5 = 15. (Check: 81 + 144 = 225 = 15².)
4. A rectangle measures 5 by 12. Find the length of its diagonal.
The diagonal is the hypotenuse of a right triangle with legs 5 and 12: √(25 + 144) = √169 = 13.

✏️ Practice — try these, take hints as needed

1. A right triangle has legs 3 and 4. Find the hypotenuse.
a² + b² = c².
9 + 16 = 25.
√25.
5
2. A right triangle has a hypotenuse of 17 and a leg of 8. Find the other leg.
8² + b² = 17².
64 + b² = 289.
b² = 225.
15
3. A right triangle has legs 8 and 15. Find the hypotenuse.
Recognise the triple.
8-15-17.
Check 64 + 225 = 289.
17
4. A right triangle has two legs each of length 6. Find the hypotenuse in exact form.
6² + 6² = 72.
√72 = √(36 × 2).
Simplify the radical.
6√2
5. A 25-foot ladder rests against a wall with its base 7 feet from the wall. How high up the wall does it reach?
Ladder is the hypotenuse.
7² + h² = 25².
h² = 625 − 49.
24 feet

📝 Topic test — 8 questions

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