Task B-2.5. In parallelogram $A B C D$, the length of side $|A B|=8 \mathrm{~cm}$, and the length of diagonal $|A C|=8 \sqrt{3} \mathrm{~cm}$. If the angle $\measuredangle C A B=30^{\circ}$, calculate the area of this parallelogram.
## Solution.

Drop a perpendicular from vertex $B$ to the diagonal $A C$. Denote the foot of the perpendicular by $S$. Triangle $\triangle A B S$ is a right triangle, so $|A S|=|A B| \cos 30^{\circ},|A S|=4 \sqrt{3} \mathrm{~cm}$.

(1 point)

We conclude that point $S$ is the midpoint of the diagonals, and thus the given quadrilateral is a rhombus. (1 point) The area is

$$
P=a^{2} \sin 60^{\circ}=64 \cdot \frac{\sqrt{3}}{2}=32 \sqrt{3} \mathrm{~cm}^{2}
$$