Question

triangle abc is shown. which of these statements about triangle abc is true? the triangle inequality theorem proves that △abc is not a right triangle. the pythagorean theorem proves that △abc is a right triangle. the converse of the triangle inequality theorem proves that △abc is not a right triangle. the converse of the pythagorean theorem proves that △abc is a right triangle.

Explanation:

Step1: Recall the converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem states that if \(a^{2}+b^{2}=c^{2}\) in a triangle with side lengths \(a\), \(b\), and \(c\) (\(c\) is the longest side), then the triangle is a right - triangle.
Here, \(a = 9\), \(b = 12\), \(c = 15\)

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

\(a^{2}+b^{2}=9^{2}+12^{2}=81 + 144=225\)
\(c^{2}=15^{2}=225\)
Since \(a^{2}+b^{2}=c^{2}\)

Answer:

The converse of the Pythagorean Theorem proves that \(\triangle ABC\) is a right triangle.