1. **State the problem:** We have a right triangle with a hypotenuse of length 13, one leg of length $4\sqrt{3}$, and an angle of 60° opposite side $a$. We need to find the lengths of sides $a$ and $b$. 2. **Recall the properties of a 30°-60°-90° triangle:** In such a triangle, the sides are in the ratio $1 : \sqrt{3} : 2$ corresponding to angles 30°, 60°, and 90° respectively. 3. **Identify the sides:** The hypotenuse is 13, which corresponds to the side opposite 90°. 4. **Find the side opposite 60° (side $a$):** Using the ratio, side opposite 60° is $\frac{\sqrt{3}}{2}$ times the hypotenuse. $$a = 13 \times \frac{\sqrt{3}}{2} = \frac{13\sqrt{3}}{2}$$ 5. **Find the side opposite 30° (side $b$):** Using the ratio, side opposite 30° is half the hypotenuse. $$b = \frac{13}{2} = 6.5$$ 6. **Check given leg $4\sqrt{3}$:** This must correspond to side $a$ or $b$. Since $a = \frac{13\sqrt{3}}{2} \approx 11.26$ and $4\sqrt{3} \approx 6.93$, the given leg $4\sqrt{3}$ corresponds to side $b$. 7. **Recalculate side $a$ using Pythagoras:** $$a = \sqrt{13^2 - (4\sqrt{3})^2} = \sqrt{169 - 48} = \sqrt{121} = 11$$ 8. **Final answers:** - Length of $a$ is 11. - Length of $b$ is $4\sqrt{3}$ (already simplified).