What does the Pythagorean theorem state?

The Pythagorean theorem specifically states that in a right 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 expressed mathematically as c² = a² + b², where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

This theorem is fundamental in geometry because it establishes a crucial relationship between the sides of right triangles, allowing for the calculation of one side length if the lengths of the other two sides are known. The theorem is widely used in various applications, ranging from architecture to navigation and is essential for proving many other geometric concepts.