1. **Problem statement:** We have an equilateral triangle with side length $x$ and a right triangle inside it where one leg is $\sqrt{10}$ and the other leg is $x$. We need to find $x$ in simplest radical form with a rational denominator. 2. **Key fact:** In an equilateral triangle, all sides are equal and all angles are $60^\circ$. 3. **Using the right triangle:** The right triangle inside splits the equilateral triangle, so the angle adjacent to $x$ is $30^\circ$ or $60^\circ$ (since the equilateral triangle angles are $60^\circ$). 4. **Using trigonometry:** If the side adjacent to the right angle is $x$ and the opposite side is $\sqrt{10}$, then by definition of tangent, $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{10}}{x}$$ Since the angle is $30^\circ$ or $60^\circ$, check which fits: $$\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577$$ $$\tan(60^\circ) = \sqrt{3} \approx 1.732$$ Calculate $\frac{\sqrt{10}}{x}$ for each to find $x$. 5. **Assuming angle is $30^\circ$:** $$\frac{\sqrt{10}}{x} = \frac{1}{\sqrt{3}}$$ Multiply both sides by $x$ and then by $\sqrt{3}$: $$\sqrt{10} \sqrt{3} = x$$ $$x = \sqrt{30}$$ 6. **Rationalize denominator check:** $x = \sqrt{30}$ is already in simplest radical form with no denominator. 7. **Answer:** The length of side $x$ is $$\boxed{\sqrt{30}}$$