Question

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

Explanation:

Step1: Apply Pythagorean theorem

Let the sides of the right - triangle be $a = 6$, $b$ (the unknown side), and $c=3\sqrt{5}$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$ or $c^{2}-a^{2}=b^{2}$.

Step2: Substitute values

Substitute $a = 6$ and $c = 3\sqrt{5}$ into $b^{2}=c^{2}-a^{2}$. We get $b^{2}=(3\sqrt{5})^{2}-6^{2}$.

Step3: Calculate squares

$(3\sqrt{5})^{2}=3^{2}\times(\sqrt{5})^{2}=9\times5 = 45$ and $6^{2}=36$. So $b^{2}=45 - 36$.

Step4: Find $b^{2}$ value

$b^{2}=45-36 = 9$.

Step5: Solve for $b$

Take the square - root of both sides. Since $b>0$ (as it represents the length of a side of a triangle), $b=\sqrt{9}=3$.

Answer:

$3$