1. **State the problem:** We have an equilateral triangle with an altitude of length 10, and we need to find the length of side $x$ in simplest radical form with a rational denominator. 2. **Recall properties of an equilateral triangle:** All sides are equal, and the altitude splits the triangle into two 30-60-90 right triangles. 3. **Use the 30-60-90 triangle ratio:** In such a triangle, the sides are in the ratio $1 : \sqrt{3} : 2$, where the shortest side is opposite the 30° angle, the side opposite 60° is $\sqrt{3}$ times the shortest side, and the hypotenuse is twice the shortest side. 4. **Identify sides:** The altitude corresponds to the side opposite the 60° angle, so if the shortest side is $s$, then the altitude is $s\sqrt{3}$. 5. **Set up the equation:** $$10 = s\sqrt{3}$$ 6. **Solve for $s$:** $$s = \frac{10}{\sqrt{3}}$$ 7. **Rationalize the denominator:** $$s = \frac{10}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{10\sqrt{3}}{3}$$ 8. **Find the side length $x$:** The side length $x$ is twice $s$ (the hypotenuse of the 30-60-90 triangle): $$x = 2s = 2 \times \frac{10\sqrt{3}}{3} = \frac{20\sqrt{3}}{3}$$ **Final answer:** $$x = \frac{20\sqrt{3}}{3}$$