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 \(1:1:\sqrt{2}\), where the hypotenuse \(c\) is \(\sqrt{2}\) times the length of each leg (\(a = b\)). Let the length of the leg be \(x\) and the hypotenuse \(c = 12\).

Step2: Use the side - ratio formula

The formula for the hypotenuse of a 45 - 45 - 90 triangle is \(c=\sqrt{2}x\). Substitute \(c = 12\) into the formula: \(12=\sqrt{2}x\).

Step3: Solve for \(x\)

To solve for \(x\), we have \(x=\frac{12}{\sqrt{2}}\). Rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{2}\): \(x=\frac{12\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{12\sqrt{2}}{2}=6\sqrt{2}\).

Answer:

\(6\sqrt{2}\)