Question

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

Explanation:

Step1: Apply Pythagorean theorem

Let the third - side be $x$. For a right - triangle with sides $a = 4$, $b=x$, and hypotenuse $c = 4\sqrt{5}$, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, so $4^{2}+x^{2}=(4\sqrt{5})^{2}$.

Step2: Expand the equations

$16 + x^{2}=16\times5$.

Step3: Simplify the equation

$x^{2}=80 - 16$.

Step4: Calculate the value of $x^{2}$

$x^{2}=64$.

Step5: Solve for $x$

$x=\sqrt{64}=8$.

Answer:

$8$