Question

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

Explanation:

Step1: Apply Pythagorean theorem

Let the unknown side be \(x\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 9\) (hypotenuse) and \(b=3\sqrt{5}\). So \(x^{2}+(3\sqrt{5})^{2}=9^{2}\).

Step2: Simplify the equation

Expand \((3\sqrt{5})^{2}=9\times5 = 45\) and \(9^{2}=81\). The equation becomes \(x^{2}+45 = 81\). Then \(x^{2}=81 - 45\).

Step3: Solve for \(x\)

\(x^{2}=36\), so \(x=\sqrt{36}=6\).

Answer:

\(6\)