8. Famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with diameter $P Q$. Point $M$ is the midpoint of the skateboard, $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take if it is known that the angular measure of the arc $A B$ is $24^{\circ} ?$

![](https://cdn.mathpix.com/cropped/2024_05_06_77078573ff69ec52d722g-4.jpg?height=284&width=496&top_left_y=1682&top_left_x=543)
Answer: $12^{\circ}$.

Solution: Extend the line $A C$ to intersect the circle at point $D$ (see figure). The chord $A D$ is perpendicular to the diameter $P Q$, therefore, it is bisected by it. Thus, $C M$ is the midline of triangle $A B D$, so $C M \| B D$ and, therefore, $\angle A C M=\angle A D B$. The angle $\angle A D B$ is an inscribed angle, subtending the arc $A B$, hence it is equal to half of the arc.

![](https://cdn.mathpix.com/cropped/2024_05_06_77078573ff69ec52d722g-5.jpg?height=463&width=499&top_left_y=430&top_left_x=542)