Which statement about Pythagorean triples is true for the pairs 3-4-5 and 5-12-13?

Which statement about Pythagorean triples is true for the pairs 3-4-5 and 5-12-13?

• A. Only 3-4-5 is a Pythagorean triple
• B. Only 5-12-13 is a Pythagorean triple
• C. Neither
• D. Both 3-4-5 and 5-12-13 are Pythagorean triples

Answer: (D)

Pythagorean triples are integer triples that satisfy a^2 + b^2 = c^2, which describes a right triangle with integer side lengths. For 3-4-5, 3^2 + 4^2 = 9 + 16 = 25, and 25 = 5^2, so it fits the relation. For 5-12-13, 5^2 + 12^2 = 25 + 144 = 169, and 169 = 13^2, so this pair also fits. Since each set satisfies the Pythagorean equation, both are Pythagorean triples. These particular triples are primitive as well, meaning they aren’t multiples of a smaller triple. Therefore, both pairs form Pythagorean triples.

Pythagorean triples are integer triples that satisfy a^2 + b^2 = c^2, which describes a right triangle with integer side lengths. For 3-4-5, 3^2 + 4^2 = 9 + 16 = 25, and 25 = 5^2, so it fits the relation. For 5-12-13, 5^2 + 12^2 = 25 + 144 = 169, and 169 = 13^2, so this pair also fits. Since each set satisfies the Pythagorean equation, both are Pythagorean triples. These particular triples are primitive as well, meaning they aren’t multiples of a smaller triple. Therefore, both pairs form Pythagorean triples.