Question

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

Explanation:

Step1: Identify triangle type (right triangle)

This is a right - triangle, so we use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse, and \(a,b\) are the legs. Here, the hypotenuse \(c = 2\sqrt{13}\), one leg \(a = 6\), and we need to find the other leg \(b\). Rearranging the Pythagorean theorem for \(b\), we get \(b=\sqrt{c^{2}-a^{2}}\) (since \(c\) is the hypotenuse, \(c>a\) as \(2\sqrt{13}=\sqrt{4\times13}=\sqrt{52}\) and \(6 = \sqrt{36}\), and \(\sqrt{52}>\sqrt{36}\)).

Step2: Substitute values into the formula

Substitute \(c = 2\sqrt{13}\) and \(a = 6\) into the formula \(b=\sqrt{c^{2}-a^{2}}\). First, calculate \(c^{2}=(2\sqrt{13})^{2}=2^{2}\times(\sqrt{13})^{2}=4\times13 = 52\), and \(a^{2}=6^{2}=36\). Then \(c^{2}-a^{2}=52 - 36=16\).

Step3: Calculate the square root

Now, \(b=\sqrt{16}=4\).

Answer:

4