What relationship does the Pythagorean theorem establish?

The Pythagorean theorem establishes a fundamental relationship between the lengths of the sides of a right triangle. Specifically, it 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 can be represented mathematically as ( a^2 + b^2 = c^2 ), where ( c ) is the length of the hypotenuse and ( a ) and ( b ) are the lengths of the other two sides.

Understanding this theorem is essential in various applications, such as determining distances and in fields such as physics and engineering. The theorem also serves as a foundation for more advanced mathematics and can be applied to solve problems involving right triangles in real-world situations, such as construction, navigation, and computer graphics.