Question

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

Explanation:

Step1: Identify the triangle type

The triangle is a right - isosceles triangle (since two angles are $45^{\circ}$ and one is $90^{\circ}$), so the legs are equal, and the hypotenuse $h$ and leg $l$ have the relationship $h = l\sqrt{2}$.

Step2: Apply the formula for the leg length

We know the hypotenuse length is $12$. Let the leg length be $x$. From $h = x\sqrt{2}$, we can solve for $x$: $x=\frac{h}{\sqrt{2}}$. Substitute $h = 12$ into the formula, we get $x=\frac{12}{\sqrt{2}}$.

Step3: Rationalize the denominator

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$: $x=\frac{12\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{12\sqrt{2}}{2}=6\sqrt{2}$.

Answer:

$6\sqrt{2}$