9.2 Points A, B, C, D are consecutive vertices of a regular n-gon. What is n if $\angle \mathrm{ACD}=120^{\circ}$?
Solution. We consider the given n-gon as inscribed.

![](https://cdn.mathpix.com/cropped/2024_05_06_3c0da93b6fd234c4e438g-1.jpg?height=235&width=511&top_left_y=1296&top_left_x=607)

Each small arc between adjacent vertices is $\frac{1}{\mathrm{n}} \cdot 360^{\circ}$. The angle $\mathrm{ACD}$ is an inscribed angle and subtends an arc consisting of $\mathrm{n}-3$ small arcs, i.e., an arc of $(\mathrm{n}-3) \cdot \frac{1}{\mathrm{n}} \cdot 360^{\circ}$. Therefore, $\angle \mathrm{ACD}=\frac{1}{2} \cdot \frac{\mathrm{n}-3}{\mathrm{n}} \cdot 360^{\circ}=\frac{\mathrm{n}-3}{\mathrm{n}} \cdot 180^{\circ}$. According to the problem, $\frac{\mathrm{n}-3}{\mathrm{n}} \cdot 180^{\circ}=240^{\circ}$. Solving for $\mathrm{n}$, we get $\mathrm{n}=9$.