What does the Pythagorean theorem state for a right-angled triangle?

The Pythagorean theorem specifically 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 can be mathematically represented as a² + b² = c², where c is the hypotenuse and a and b are the other two sides. This foundational principle is crucial in geometry and is widely used in various applications, including physics, engineering, and architecture, to determine distances and relationships between different parts of right-angled triangles. Thus, a² + b² = c² effectively summarizes this essential concept.