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

Explanation:

Step1: Assume it's a 45 - 45 - 90 triangle

In a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$. Let the legs be of length $x$ and the hypotenuse be $c$. The Pythagorean theorem for a right - triangle is $a^{2}+b^{2}=c^{2}$. In a 45 - 45 - 90 triangle with legs $x$ and hypotenuse $c$, we have $x^{2}+x^{2}=c^{2}$. Since $c = \sqrt{3}$, then $2x^{2}=(\sqrt{3})^{2}$.

Step2: Solve the equation for $x$

We have $2x^{2}=3$. Then $x^{2}=\frac{3}{2}$. Taking the square root of both sides, $x=\sqrt{\frac{3}{2}}$.

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{2}$. So $x=\frac{\sqrt{3}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{6}}{2}$.

Answer:

$\frac{\sqrt{6}}{2}$