43. In triangle \( A B C \), the measure of \( \angle A \) is \( 60^{\circ} \) and the measure of \( \angle \) is \( 30^{\circ} \). If \( \overline{A B} \) is 8 units long, what is the area, in square units, of triangle \( A B C \) ? A. 4 B. \( 4 \sqrt{3} \) C. 8 D. \( 8 \sqrt{3} \) E. \( 16 \sqrt{3} \)

To find the area of triangle \(ABC\), we can use the formula for the area of a triangle given two sides and the included angle. The formula is: \[ \text{Area} = \frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\text{angle between the sides}) \] Given: - \( \angle A = 60^{\circ} \) - \( \angle B = 30^{\circ} \) - \( \overline{AB} = 8 \) units We can calculate the area of triangle \(ABC\) using the formula above. Let's calculate it step by step. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\times 8\times 8\sin\left(60\right)\) - step1: Multiply the terms: \(32\sin\left(60\right)\) The area of triangle \(ABC\) is approximately \(32 \sin(60^\circ)\) square units. Now, we need to calculate the value of \(32 \sin(60^\circ)\) to find the exact area. Let's calculate it. Calculate the value by following steps: - step0: Calculate: \(32\sin\left(60\right)\) The area of triangle \(ABC\) is approximately \(32 \sin(60^\circ) \approx 32 \times \frac{\sqrt{3}}{2} \approx 16\sqrt{3}\) square units. Therefore, the correct answer is E. \(16\sqrt{3}\).