1. **State the problem:** We have a right triangle with angles 30°, 60°, and 90°. The side opposite 30° is given as $22\sqrt{3}$, and we need to find the lengths of sides $x$ (hypotenuse) and $y$ (side opposite 60°). 2. **Recall the properties of a 30°-60°-90° triangle:** - The side opposite 30° is $\frac{1}{2}$ the hypotenuse. - The side opposite 60° is $\frac{\sqrt{3}}{2}$ times the hypotenuse. 3. **Set up the relationships:** Let the hypotenuse be $x$. Then the side opposite 30° is $\frac{x}{2}$, and the side opposite 60° is $\frac{\sqrt{3}}{2}x$. 4. **Use the given side opposite 30°:** $$22\sqrt{3} = \frac{x}{2}$$ Multiply both sides by 2: $$2 \times 22\sqrt{3} = \cancel{2} \times \frac{x}{\cancel{2}}$$ $$44\sqrt{3} = x$$ 5. **Find side $y$ opposite 60°:** $$y = \frac{\sqrt{3}}{2} x = \frac{\sqrt{3}}{2} \times 44\sqrt{3}$$ Simplify inside the multiplication: $$y = 44 \times \frac{\sqrt{3} \times \sqrt{3}}{2} = 44 \times \frac{3}{2}$$ $$y = 44 \times 1.5 = 66$$ **Final answers:** $$x = 44\sqrt{3}$$ $$y = 66$$