Question

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

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\). Let one leg \(a = 4\), the hypotenuse \(c=4\sqrt{5}\). We need to find the other leg \(b\).
Substitute into the formula: \(4^{2}+b^{2}=(4\sqrt{5})^{2}\)

Step2: Simplify the equation

Calculate \(4^{2}=16\) and \((4\sqrt{5})^{2}=16\times5 = 80\).
The equation becomes \(16 + b^{2}=80\).

Step3: Solve for \(b^{2}\)

Subtract 16 from both sides: \(b^{2}=80 - 16=64\).

Step4: Find \(b\)

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

Answer:

\(8\)