1. **Problem statement:** We have an isosceles right triangle with two equal sides each of length $x$ and the base side length is 1. We need to find the length of side $x$ in simplest radical form with a rational denominator. 2. **Formula and rules:** In an isosceles right triangle, the legs are equal and the hypotenuse is related to the legs by the Pythagorean theorem: $$\text{hypotenuse}^2 = \text{leg}^2 + \text{leg}^2$$ Since the base side is 1 and it is the hypotenuse, we have: $$1^2 = x^2 + x^2$$ 3. **Intermediate work:** $$1 = 2x^2$$ Divide both sides by 2: $$\frac{1}{\cancel{2}} = \frac{2x^2}{\cancel{2}} \Rightarrow \frac{1}{2} = x^2$$ 4. **Solve for $x$:** $$x = \sqrt{\frac{1}{2}}$$ 5. **Rationalize the denominator:** $$x = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$ 6. **Final answer:** $$x = \frac{\sqrt{2}}{2}$$ This is the length of side $x$ in simplest radical form with a rational denominator.