Question

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

Explanation:

Step1: Identify 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 $c = a\sqrt{2}$ (or $c = b\sqrt{2}$). Let the length of each leg be $x$. The hypotenuse $c=\sqrt{6}$.

Step2: Set up equation

We know that $c = x\sqrt{2}$, so $\sqrt{6}=x\sqrt{2}$.

Step3: Solve for $x$

To solve for $x$, we divide both sides of the equation $\sqrt{6}=x\sqrt{2}$ by $\sqrt{2}$: $x=\frac{\sqrt{6}}{\sqrt{2}}$. Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$: $x=\frac{\sqrt{6}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{12}}{2}=\frac{2\sqrt{3}}{2}=\sqrt{3}$.

Answer:

$\sqrt{3}$