Which of the following sets of side lengths forms a ${\mathrm{Pythagorean\; triple}}^{ }$?

• A. $11,60,61$
• B. $10,24,29$
• C. $2,3,13$
• D. $5,7,12$

Answer: (B)

Verified step by step guidance

1. Recall that a Pythagorean triple consists of three positive integers \(a\), \(b\), and \(c\) such that \(a^2 + b^2 = c^2\), where \(c\) is the longest side (hypotenuse).
2. For each set of side lengths, first identify the largest number as the potential hypotenuse \(c\), and the other two as legs \(a\) and \(b\).
3. Calculate \(a^2 + b^2\) for the two smaller sides and compare it to \(c^2\) for the largest side.
4. If \(a^2 + b^2 = c^2\) holds true exactly, then the set forms a Pythagorean triple; otherwise, it does not.
5. Apply this test to each given set: (11, 60, 61), (10, 24, 29), (2, 3, 13), and (5, 7, 12) to determine which one satisfies the Pythagorean theorem.