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 \(x\). According to Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), here \(c = 8\), \(a=4\sqrt{3}\), and \(b=x\). So \(x^{2}+(4\sqrt{3})^{2}=8^{2}\).

Step2: Simplify the equation

First, calculate \((4\sqrt{3})^{2}=4^{2}\times(\sqrt{3})^{2}=16\times3 = 48\) and \(8^{2}=64\). The equation becomes \(x^{2}+48 = 64\). Then \(x^{2}=64 - 48\).

Step3: Solve for \(x\)

\(x^{2}=16\), so \(x=\sqrt{16}\).

Answer:

\(4\)