1. **Stating the problem:** Given a triangle with an angle of 60° and an adjacent side of length $52\sqrt{3}$, find the missing sides $x$ and $y$ of a right triangle with angles 30° and 60°, where one side is $5\sqrt{3}$. 2. **Relevant formulas and rules:** For a 30°-60°-90° right triangle, the sides have a fixed ratio: $$\text{Opposite 30°} : \text{Opposite 60°} : \text{Hypotenuse} = 1 : \sqrt{3} : 2$$ This means if the side opposite 30° is $a$, then the side opposite 60° is $a\sqrt{3}$ and the hypotenuse is $2a$. 3. **Using the given side $5\sqrt{3}$:** Since $5\sqrt{3}$ is given, identify which side it corresponds to. If $5\sqrt{3}$ is opposite 60°, then: $$x = \text{side opposite 30°} = \frac{5\sqrt{3}}{\sqrt{3}} = 5$$ 4. **Calculate the hypotenuse $y$:** Using the ratio, hypotenuse $y = 2 \times x = 2 \times 5 = 10$ 5. **Summary of sides:** $$x = 5$$ $$y = 10$$ 6. **Verification:** The side opposite 60° is $x\sqrt{3} = 5\sqrt{3}$, which matches the given side. **Final answer:** $$x = 5, \quad y = 10$$