What mathematical theorem is used in right-angled triangles?

The Pythagorean Theorem is the fundamental principle associated with right-angled triangles. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is mathematically expressed as (c^2 = a^2 + b^2), where (c) represents the length of the hypotenuse, while (a) and (b) denote the lengths of the other two sides.

This theorem is crucial in various applications, such as calculating distances, determining the lengths of triangle sides, and solving real-world problems involving right angles. Its simplicity and effectiveness in categorizing relationships between the sides of right-angled triangles make it a cornerstone of geometry and trigonometry.