Question

find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\). Let the legs be of length \(a\) and the hypotenuse be \(c\). The formula is \(c = a\sqrt{2}\). Here, one leg \(a = 6\), and we need to find the hypotenuse \(x\).

Step2: Apply the 45 - 45 - 90 triangle formula

Using \(c=a\sqrt{2}\), substitute \(a = 6\). So \(x=6\sqrt{2}\). The denominator is already rational (since \(\sqrt{2}\) is in the numerator and there is no denominator in the form of \(\frac{1}{\sqrt{2}}\) etc. after substitution).

Answer:

\(6\sqrt{2}\)