Problem 9.6. Given an obtuse triangle $ABC$ with an obtuse angle $C$. On its sides $AB$ and $BC$, points $P$ and $Q$ are marked such that $\angle ACP = CPQ = 90^\circ$. Find the length of the segment $PQ$, if it is known that $AC = 25, CP = 20, \angle APC = \angle A + \angle B$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a324d57f9e21fe03fd09g-3.jpg?height=397&width=679&top_left_y=730&top_left_x=371)
Answer: 16.

Solution. Since $\angle P C B+\angle P B C=\angle A P C=\angle P A C+\angle P B C$, we obtain $\angle P C B=\angle P A C$. Note that the right triangles $P A C$ and $Q C P$ are similar by the acute angle, and

$$
\frac{25}{20}=\frac{A C}{C P}=\frac{P C}{P Q}=\frac{20}{P Q}
$$

from which we find $P Q=\frac{20 \cdot 20}{25}=16$.