1. The problem is to find the missing sides $x$ and $y$ in the right triangle with a given side 10 and an angle of 60° without using the Pythagorean theorem. 2. We use trigonometric ratios for right triangles: - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ 3. Here, the hypotenuse is 10, the angle is 60°, $x$ is adjacent to the angle, and $y$ is opposite. 4. Calculate $x$ using cosine: $$x = 10 \times \cos(60^\circ)$$ Since $\cos(60^\circ) = \frac{1}{2}$, $$x = 10 \times \frac{1}{2} = 5$$ 5. Calculate $y$ using sine: $$y = 10 \times \sin(60^\circ)$$ Since $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, $$y = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}$$ 6. Final answers: $$x = 5$$ $$y = 5\sqrt{3}$$