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 45 - 45 - 90 right - triangle (since the non - hypotenuse sides are equal). 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 $b$ (where $a = b$) by $c=\sqrt{2}a$.
Let the length of each leg be $x$. The hypotenuse is $\sqrt{6}$.

Step2: Set up the equation

We know that for a 45 - 45 - 90 triangle, if the hypotenuse $c=\sqrt{6}$ and $c = \sqrt{2}x$ (where $x$ is the length of a leg), then we can solve for $x$ from the equation $\sqrt{6}=\sqrt{2}x$.

Step3: Solve for $x$

Divide both sides of the equation $\sqrt{6}=\sqrt{2}x$ 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:

$\sqrt{3}$