Question

question find the length of the third - side. if necessary, write in simplest radical form answer attempt 1 out of 2

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Let \(a = 4\) and \(b\) be the unknown side, and \(c=4\sqrt{5}\).

Step2: Substitute values into the formula

\(4^{2}+b^{2}=(4\sqrt{5})^{2}\).

Step3: Simplify the equation

First, calculate \(4^{2}=16\) and \((4\sqrt{5})^{2}=4^{2}\times(\sqrt{5})^{2}=16\times5 = 80\). So the equation becomes \(16 + b^{2}=80\).

Step4: Solve for \(b^{2}\)

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

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), \(b=\sqrt{64}=8\).

Answer:

8