Question

​the exterior angle of a regular polygon is 1/5 of its interior angle. how many sides does the polygon have

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{n}\mathrm{be}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{sides}\mathrm{of}\mathrm{regular}\mathrm{polygon}. \\ \mathrm{Measure}\mathrm{of}\mathrm{each}\mathrm{exterior}\mathrm{angle}=\frac{360°}{\mathrm{n}} \\ \mathrm{Measure}\mathrm{of}\mathrm{its}\mathrm{interior}\mathrm{angle}=\frac{\left(\mathrm{n}-2\right)\times 180°}{\mathrm{n}} \\ \mathrm{Now},\mathrm{according}\mathrm{to}\mathrm{question}: \\ \frac{360°}{\mathrm{n}}=\frac{1}{5}\times \frac{\left(\mathrm{n}-2\right)\times 180°}{\mathrm{n}} \\ \Rightarrow 360°=\frac{\left(\mathrm{n}-2\right)\times 180°}{5} \\ \Rightarrow 10=\mathrm{n}-2 \\ \Rightarrow \mathrm{n}=10+2 \\ \Rightarrow \boxed{\mathbf{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{12}} \end{gathered}$$