Question

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

Explanation:

Step1: Identify the triangle type

This is a right - triangle with two legs of length \(a = 8\) and \(b = 8\). We can use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\) (where \(c\) is the hypotenuse, the third side we want to find) to solve for the length of the hypotenuse.

Step2: Apply the Pythagorean theorem

Substitute \(a = 8\) and \(b = 8\) into the formula \(c^{2}=a^{2}+b^{2}\). So \(c^{2}=8^{2}+8^{2}\). Calculate \(8^{2}=64\), then \(c^{2}=64 + 64=128\).

Step3: Simplify the radical

We know that \(c=\sqrt{128}\). Factor \(128\) as \(64\times2\), so \(\sqrt{128}=\sqrt{64\times2}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we get \(\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}=8\sqrt{2}\).

Answer:

\(8\sqrt{2}\)