Question

which best explains whether a triangle with side lengths 5 cm, 13 cm, and 12 cm is a right triangle?
the triangle is a right triangle because ( 5^{2}+12^{2}=13^{2} ).
the triangle is a right triangle because ( 5 + 13>12 ).
the triangle is not a right triangle because ( 5^{2}+13^{2}>12^{2} ).
the triangle is not a right triangle because ( 5 + 12>13 ).

Explanation:

Step1: Recall the Pythagorean theorem

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

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

Here \(a = 5\), \(b=12\), \(c = 13\).
\(a^{2}+b^{2}=5^{2}+12^{2}=25 + 144=169\)
\(c^{2}=13^{2}=169\)

Answer:

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