Question

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

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). Let \(a = 6\), \(b\) be the unknown side, and \(c=2\sqrt{13}\). Then \(b^{2}=c^{2}-a^{2}\).

Step2: Substitute values

Substitute \(a = 6\) and \(c = 2\sqrt{13}\) into the formula: \(b^{2}=(2\sqrt{13})^{2}-6^{2}\).
Calculate \((2\sqrt{13})^{2}=4\times13 = 52\) and \(6^{2}=36\). So \(b^{2}=52 - 36\).

Step3: Simplify

\(b^{2}=16\). Then \(b=\sqrt{16}\).

Answer:

\(4\)