Question

find the length of the third side. if necessary, write in simplest radical form.

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem, which states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\). Let the hypotenuse \(c = 9\) and one leg \(a=\sqrt{65}\), and we want to find the other leg \(b\).

Step2: Apply the Pythagorean theorem

From \(a^{2}+b^{2}=c^{2}\), we can solve for \(b\): \(b=\sqrt{c^{2}-a^{2}}\)
Substitute \(c = 9\) and \(a=\sqrt{65}\) into the formula:
First, calculate \(c^{2}=9^{2}=81\) and \(a^{2}=(\sqrt{65})^{2}=65\)
Then \(b=\sqrt{81 - 65}=\sqrt{16}\)

Step3: Simplify the radical

\(\sqrt{16}=4\)

Answer:

\(4\)