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

Answer

Explanation:

Step1: Apply Pythagorean theorem

Let the sides of the right - triangle be $a = \sqrt{39}$, $c = 8$, and the unknown side be $b$. According to the Pythagorean theorem $c^{2}=a^{2}+b^{2}$ (where $c$ is the hypotenuse). We need to find $b$, so $b^{2}=c^{2}-a^{2}$.

Step2: Substitute values

Substitute $a = \sqrt{39}$ and $c = 8$ into the formula $b^{2}=c^{2}-a^{2}$. Then $b^{2}=8^{2}-\left(\sqrt{39}\right)^{2}=64 - 39$.

Step3: Calculate $b^{2}$

$b^{2}=64 - 39=25$.

Step4: Solve for $b$

Take the square - root of both sides. Since $b>0$ (as it represents the length of a side of a triangle), $b=\sqrt{25}=5$.