Question

question find the length of the third side. if necessary, write in simplest radical form. answer attempt 2 out of 2 4√5 8

Explanation:

Step1: Identify 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 \(b = 8\). We need to find the third side. We don't know if the third side is the hypotenuse or a leg. First, assume the third side \(x\) is the hypotenuse. Then \((4\sqrt{5})^{2}+8^{2}=x^{2}\).

Step2: Calculate squares of given sides

\((4\sqrt{5})^{2}=4^{2}\times(\sqrt{5})^{2}=16\times5 = 80\) and \(8^{2}=64\).

Step3: Solve for \(x\)

\(x^{2}=80 + 64=144\), so \(x=\sqrt{144}=12\). Now assume the third side is a leg. Let the hypotenuse be \(4\sqrt{5}\) (since \(4\sqrt{5}\approx4\times2.24 = 8.96>8\)). Then \((4\sqrt{5})^{2}-8^{2}=x^{2}\). \((4\sqrt{5})^{2}=80\) and \(8^{2}=64\), so \(x^{2}=80 - 64 = 16\), and \(x = 4\). But since \(4\sqrt{5}\) is likely the hypotenuse (as it is the longest - looking side among the given values), the third side is \(12\).

Answer:

12