If $z$ and $\bar{z}$ represent adjacent vertices of a regular polygon of $n$ sides with centre at origin and if $\frac{Im z}{Re z}=\sqrt{2}-1$, then find the value of $n$.
Could someone give me hint for this question? I am not able to use the information $\frac{Im z}{Re z}=\sqrt{2}-1$.
Best Answer
If $z$ and $\overline z$ are adjacent vertices of a regular polygon then $\angle zO\overline z$ equals $2\pi/n$ where $n$ is the number of sides of the polygon. Then if you look at $Im z/Re z$ you'll find it as a ratio in a triangle with vertices $z,O$ and the projection of $z$ on the $x$-axis. This will help you find the value of $2\pi/n$ and thus find $n$ in the end.