Why are 3 and 4 and 5 and 5 and 12 and 13 are called Pythagorean triples?

The pairs 3 and 4, 5 and 5, and 12 and 13 are called Pythagorean triples because they satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. For example, for the triple (3, 4, 5), (3^2 + 4^2 = 9 + 16 = 25), which equals (5^2). Similarly, the other pairs can be verified, confirming they also meet this condition.