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, so we can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side), and \(a\) and \(b\) are the legs. Here, \(c = \sqrt{73}\) and one leg \(a = 8\), we need to find the other leg \(b\).

Step2: Apply the Pythagorean theorem

From \(a^{2}+b^{2}=c^{2}\), we can re - arrange it to solve for \(b\): \(b^{2}=c^{2}-a^{2}\)

Substitute \(a = 8\) and \(c=\sqrt{73}\) into the formula:

\(b^{2}=(\sqrt{73})^{2}-8^{2}\)

We know that \((\sqrt{x})^{2}=x\) for \(x\geq0\), so \((\sqrt{73})^{2}=73\) and \(8^{2} = 64\)

Then \(b^{2}=73 - 64=9\)

Step3: Solve for \(b\)

Take the square root of both sides: \(b=\sqrt{9} = 3\) (we take the positive square root because the length of a side of a triangle is positive)

Answer:

The length of the third side is \(3\)