1. **Problem statement:** Find the length of side $x$ in simplest radical form with a rational denominator in a right isosceles triangle with angles $45^\circ$, $45^\circ$, and $90^\circ$, where the hypotenuse is 1. 2. **Formula and rules:** In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, the sides opposite the $45^\circ$ angles are equal, and the hypotenuse is $\sqrt{2}$ times the length of each leg. 3. **Set up the equation:** Let each leg be $x$. Then the hypotenuse $h$ satisfies: $$h = x \sqrt{2}$$ Given $h = 1$, we have: $$1 = x \sqrt{2}$$ 4. **Solve for $x$:** $$x = \frac{1}{\sqrt{2}}$$ 5. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{2}$: $$x = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$ 6. **Final answer:** $$x = \frac{\sqrt{2}}{2}$$ This is the length of side $x$ in simplest radical form with a rational denominator.