Question

which best explains whether a triangle with side lengths 5 cm, 13 cm, and 12 cm is a right triangle? the triangle is not a right triangle because 5² + 13² > 12². the triangle is a right triangle because 5 + 13 > 12. the triangle is a right triangle because 5² + 12² = 13². the triangle is not a right triangle because 5 + 12 > 13.

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with side lengths \(a\), \(b\), and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Here, the longest side is \(c = 13\), and \(a = 5\), \(b = 12\).

Step2: Calculate \(a^{2}+b^{2}\) and \(c^{2}\)

\(a^{2}=5^{2}=25\), \(b^{2}=12^{2}=144\), so \(a^{2}+b^{2}=25 + 144=169\). And \(c^{2}=13^{2}=169\). Since \(5^{2}+12^{2}=13^{2}\), the triangle satisfies the Pythagorean theorem.

Answer:

The triangle is a right triangle because \(5^{2}+12^{2}=13^{2}\).