How do Pythagorean triples relate to right triangles?

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, which states that for a right triangle with legs of lengths a and b and a hypotenuse of length c, the relationship $a^2+b^2=c^2$ holds true.12

While the sides of a right triangle often result in irrational numbers when two sides are chosen at random, Pythagorean triples represent the specific cases where all three side lengths are integers. For example, the set (3, 4, 5) is a Pythagorean triple because $3^2+4^2=5^2$ (or $9+16=25$), meaning these lengths can perfectly form the sides of a right triangle. Ancient mathematicians, including the Babylonians, identified these triples to solve geometric problems involving right triangles long before the formalization of the theorem by the Greeks.21