1. **State the problem:** We have a right triangle with a 60° angle, the side opposite this angle is 12, the adjacent side is $x$, and the hypotenuse is $y$. We need to find $x$ and $y$ in simplest radical form. 2. **Recall special right triangle rules:** In a 30°-60°-90° triangle, the sides are in the ratio $1 : \sqrt{3} : 2$ corresponding to angles $30^\circ : 60^\circ : 90^\circ$ respectively. 3. **Identify sides relative to 60° angle:** Opposite 60° is $12$, adjacent to 60° is $x$, hypotenuse is $y$. 4. **Use ratio to find $x$:** The side opposite 60° corresponds to $\sqrt{3}$ times the shortest side (opposite 30°). Let the shortest side be $s$. Then: $$12 = s \sqrt{3} \implies s = \frac{12}{\sqrt{3}}$$ 5. **Simplify $s$:** $$s = \frac{12}{\sqrt{3}} = \frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12 \sqrt{3}}{3} = 4 \sqrt{3}$$ 6. **Find $x$ (adjacent to 60°, opposite 30°):** $$x = s = 4 \sqrt{3}$$ 7. **Find $y$ (hypotenuse):** Hypotenuse is twice the shortest side: $$y = 2s = 2 \times 4 \sqrt{3} = 8 \sqrt{3}$$ **Final answers:** $$x = 4 \sqrt{3}$$ $$y = 8 \sqrt{3}$$