1. **State the problem:** We have a 30°-60°-90° right triangle with hypotenuse length 10. We need to find the lengths of sides $x$ (opposite 60°) and $y$ (opposite 30°). 2. **Recall the properties of a 30°-60°-90 triangle:** The sides are in the ratio $1 : \sqrt{3} : 2$, where the side opposite 30° is $\frac{1}{2}$ the hypotenuse, and the side opposite 60° is $\frac{\sqrt{3}}{2}$ times the hypotenuse. 3. **Calculate $y$ (opposite 30°):** $$y = \frac{1}{2} \times 10 = 5$$ 4. **Calculate $x$ (opposite 60°):** $$x = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3}$$ 5. **Final answers:** $$x = 5\sqrt{3}$$ $$y = 5$$ These values satisfy the triangle side ratios and the Pythagorean theorem.