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 legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\). Here, we know the hypotenuse \(c = 9\) and one leg \(a=6\), and we need to find the other leg \(b\). Rearranging the formula for \(b\), we get \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute the values

Substitute \(c = 9\) and \(a = 6\) into the formula: \(b=\sqrt{9^{2}-6^{2}}=\sqrt{81 - 36}=\sqrt{45}\).

Step3: Simplify the radical

Simplify \(\sqrt{45}\). We can factor \(45\) as \(9\times5\), so \(\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}\).

Answer:

\(3\sqrt{5}\)