1. **State the problem:** We have a right trapezoid with the following known sides and angles: - Top base = 13 - Left vertical side = $4\sqrt{3}$ - Top slant side $a$ forms a 60° angle with the horizontal top base - Bottom base = $b$ We need to find the lengths of $a$ and $b$ in simplest radical form. 2. **Analyze the trapezoid:** The trapezoid has two right angles on the left side, so the left side is vertical. The top slant side $a$ makes a 60° angle with the horizontal top base. 3. **Use trigonometry on the right triangle formed by $a$:** The vertical side adjacent to the angle is $4\sqrt{3}$. Since $a$ forms a 60° angle with the horizontal, the vertical side corresponds to the opposite side of the 60° angle. Using sine: $$\sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4\sqrt{3}}{a}$$ 4. **Calculate $a$:** $$a = \frac{4\sqrt{3}}{\sin 60^\circ}$$ Recall that $\sin 60^\circ = \frac{\sqrt{3}}{2}$, so: $$a = \frac{4\sqrt{3}}{\frac{\sqrt{3}}{2}} = 4\sqrt{3} \times \frac{2}{\sqrt{3}}$$ Cancel $\sqrt{3}$: $$a = 4 \times \cancel{\sqrt{3}} \times \frac{2}{\cancel{\sqrt{3}}} = 4 \times 2 = 8$$ 5. **Find $b$ (bottom base):** The horizontal component of $a$ is adjacent to the 60° angle: $$\cos 60^\circ = \frac{\text{adjacent}}{a} = \frac{\text{horizontal component}}{8}$$ Calculate horizontal component: $$\text{horizontal component} = 8 \times \cos 60^\circ = 8 \times \frac{1}{2} = 4$$ The bottom base $b$ is the sum of the top base (13) plus the horizontal component (4): $$b = 13 + 4 = 17$$ **Final answers:** $$a = 8$$ $$b = 17$$