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 related to the legs $a$ and $b$ (equal in this case) by $c = a\sqrt{2}$. Let the length of each leg be $x$. The hypotenuse is $\sqrt{11}$.

Step2: Set up the equation

We have $\sqrt{11}=x\sqrt{2}$.

Step3: Solve for $x$

To isolate $x$, we divide both sides of the equation by $\sqrt{2}$: $x=\frac{\sqrt{11}}{\sqrt{2}}$.

Step4: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{2}$: $x = \frac{\sqrt{11}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{22}}{2}$.

Answer:

$\frac{\sqrt{22}}{2}$