Question

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

Explanation:

Step1: Identify the Pythagorean theorem

For a right - triangle with sides \(a\), \(b\), and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Let \(a = 4\sqrt{5}\) and \(c = 8\). We need to find \(b\). Rearranging the formula for \(b\) gives \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute the values

First, calculate \(a^{2}=(4\sqrt{5})^{2}=4^{2}\times(\sqrt{5})^{2}=16\times5 = 80\) and \(c^{2}=8^{2}=64\). Since \(c^{2}c\). Let \(a = 4\sqrt{5}\) and \(b\) be the unknown side and \(c\) be the hypotenuse. \(a^{2}=(4\sqrt{5})^{2}=80\). Let's assume the two given sides are the legs of the right - triangle. Then, by the Pythagorean theorem \(c=\sqrt{(4\sqrt{5})^{2}+8^{2}}\).

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Answer:

12