Question

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

Explanation:

Step1: Identify the triangle type

This is a right - isosceles triangle. In a right - isosceles triangle, if the length of the hypotenuse is $c$ and the length of each of the legs is $a$, by the Pythagorean theorem $c^{2}=a^{2}+a^{2}=2a^{2}$. Here $c = 7$ and $a=x$.
So, $7^{2}=2x^{2}$, which gives $49 = 2x^{2}$.

Step2: Solve for $x$

First, rewrite the equation as $x^{2}=\frac{49}{2}$. Then $x=\sqrt{\frac{49}{2}}$.

Step3: Rationalize the denominator

Multiply the numerator and denominator inside the square - root by 2: $x=\sqrt{\frac{49\times2}{2\times2}}=\frac{7\sqrt{2}}{2}$.

Answer:

$\frac{7\sqrt{2}}{2}$