The problem asks us to find the length of side $x$ in the given right triangle, expressed in simplest radical form with a rational denominator. We are given that the hypotenuse has length 1, and the triangle is isosceles, which means the two legs have equal length.

1. Problem Description

The problem asks us to find the length of side $x$ in the given right triangle, expressed in simplest radical form with a rational denominator. We are given that the hypotenuse has length 1, and the triangle is isosceles, which means the two legs have equal length.

2. Solution Steps

Let the length of the two legs be $a$. Since the triangle is isosceles and right-angled, both legs have the same length. We are given that the length of one leg is $x$, so $a=x$. By the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. Therefore, we have

$x^2 + x^2 = 1^2$

$2x^2 = 1$

$x^2 = \frac{1}{2}$

$x = \sqrt{\frac{1}{2}}$

$x = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{2}$:

$x = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

3. Final Answer

The length of side $x$ is $\frac{\sqrt{2}}{2}$.