Question

if two sides of a right triangle are 40 and 41 and the sides form a pythagorean triple, find the third side.

Explanation:

Step1: Identify Hypotenuse Possibility

In a right triangle, the hypotenuse (\(c\)) is the longest side. So we check if 41 is the hypotenuse or a leg. Let \(a = 40\), \(b\) be the unknown leg, \(c = 41\) (hypotenuse) or \(c\) (hypotenuse) with \(a = 40\), \(b = 41\).

Step2: Apply Pythagorean Theorem

Case 1: 41 is hypotenuse. Then \(a^2 + b^2 = c^2\) → \(40^2 + b^2 = 41^2\).
\(1600 + b^2 = 1681\) → \(b^2 = 1681 - 1600 = 81\) → \(b = \sqrt{81} = 9\).

Case 2: 41 is a leg. Then \(40^2 + 41^2 = c^2\) → \(1600 + 1681 = c^2\) → \(c^2 = 3281\), \(c \approx 57.28\) (not integer, but Pythagorean triple needs integers. So Case 1 is valid as 9,40,41 is a triple).

Answer:

9