Problem 1. Given a right-angled triangle $ABC$ with the right angle at vertex $C$. Let point $P$ be the midpoint of side $\overline{AB}$ and suppose $\measuredangle PCA=90^{\circ}$, $\measuredangle BCP=30^{\circ}$, $\overline{AC}=3 \text{~cm}$. Find the length of side $\overline{BC}$.
Solution. We complete triangle $ABC$ to parallelogram $AECB$. Then, $\measuredangle AEC = 30^{\circ}$, which means that triangle $AEC$ is half of some equilateral triangle, so

![](https://cdn.mathpix.com/cropped/2024_06_05_a1a5b9bf3a5d296219b1g-10.jpg?height=250&width=545&top_left_y=1728&top_left_x=934)

$$
\overline{AE} = 2 \overline{AC} = 2 \cdot 3 = 6
$$

which means $\overline{BC} = 6 \text{ cm}$.