Hypotenuse Approximation

Technique: hypotenuse from two legs

For a right triangle with legs a and b (with a the larger), the hypotenuse is √(a²+b²). A fast mental approximation is:

√(a²+b²) ≈ a + b²/(2a)

This Taylor-style estimate gets you in the right neighborhood. Every problem in this round is built from a Pythagorean triple, so the exact hypotenuse is a clean integer — type that integer.

Examples

• legs 4, 3 → 3-4-5 triple → 5
• legs 12, 5 → 5-12-13 triple → 13
• legs 24, 7 → 7-24-25 triple → 25
• legs 40, 30 → scaled 3-4-5 by 10 → 50

Why does this work?

From c = a√(1 + (b/a)²) with b/a small, the first-order expansion √(1+x) ≈ 1 + x/2 gives c ≈ a + b²/(2a). The error stays under a few percent when b ≤ a. Memorise the common triples (3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29) and their scalings — most real problems collapse to one of these.

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