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 $a = 4$, $b$ (unknown), and $c=4\sqrt{5}$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$.

Step2: Rearrange the formula to solve for $b$

$b^{2}=c^{2}-a^{2}$. Substitute $a = 4$ and $c = 4\sqrt{5}$ into the formula: $b^{2}=(4\sqrt{5})^{2}-4^{2}$.

Step3: Calculate the values

$(4\sqrt{5})^{2}=4^{2}\times(\sqrt{5})^{2}=16\times5 = 80$ and $4^{2}=16$. So $b^{2}=80 - 16=64$.

Step4: Find the value of $b$

Take the square - root of both sides: $b=\sqrt{64}=8$.

Answer:

$8$