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. The Pythagorean theorem states that for a right - triangle with hypotenuse \(c\) and legs \(a\) and \(b\), \(a^{2}+b^{2}=c^{2}\). Let the hypotenuse \(c = \sqrt{65}\) and one leg \(a = 4\), and we want to find the other leg \(b\).

Step2: Rearrange the Pythagorean theorem

We can rearrange the formula to solve for \(b\): \(b=\sqrt{c^{2}-a^{2}}\)

Step3: Substitute the values

Substitute \(c = \sqrt{65}\) and \(a = 4\) into the formula. First, calculate \(c^{2}=(\sqrt{65})^{2}=65\) and \(a^{2}=4^{2} = 16\). Then \(c^{2}-a^{2}=65 - 16=49\).

Step4: Calculate \(b\)

Then \(b=\sqrt{49}=7\)

Answer:

7