Question

In figures like octagon , nonagon and decagon the line of symmetry depends on the number of sides.
For example, Octagon - 8 sides
8 lines of symmetry
Nonagon - 9 sides
9 lines of symmetry
Decagon - 10 sides
10 lines of symmetry
Is this property true or false ?

Answer 1

$$\begin{gathered} \mathbf{Line}\mathbf{}\mathbf{of}\mathbf{}\mathbf{symmetry}\mathbf{}\mathbf{is}\mathbf{}\mathbf{a}\mathbf{}\mathbf{line}\mathbf{}\mathbf{that}\mathbf{}\mathbf{divides}\mathbf{}\mathbf{a}\mathbf{}\mathbf{figure}\mathbf{}\mathbf{into}\mathbf{}\mathbf{two}\mathbf{}\mathbf{congruent}\mathbf{}\mathbf{parts}\mathbf{,}\mathbf{} \\ \mathbf{each}\mathbf{}\mathbf{of}\mathbf{}\mathbf{which}\mathbf{}\mathbf{is}\mathbf{}\mathbf{the}\mathbf{}\mathbf{mirror}\mathbf{}\mathbf{images}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{other}. \\ \mathrm{Each}\mathrm{of}\mathrm{the}\mathrm{part}\mathrm{when}\mathrm{folded}\mathrm{coincides}\mathrm{with}\mathrm{each}\mathrm{other}. \\ \mathbf{An}\mathbf{}\mathbf{octagon}\mathbf{}\mathbf{is}\mathbf{}\mathbf{a}\mathbf{}\mathbf{polygon}\mathbf{}\mathbf{having}\mathbf{}\mathbf{8}\mathbf{}\mathbf{sides}. \\ \mathrm{In}\mathrm{case}\mathrm{the}\mathbf{octagon}\mathbf{}\mathbf{is}\mathbf{}\mathbf{an}\mathbf{}\mathbf{irregular}\mathbf{}\mathbf{figure},\mathrm{then}\mathrm{one}\mathrm{can}\mathrm{assume}\mathrm{that}\mathrm{there} \\ \mathrm{is}\mathbf{no}\mathbf{}\mathbf{lines}\mathbf{}\mathbf{of}\mathbf{}\mathbf{symmetry}. \\ \mathrm{But}\mathrm{if}\mathrm{the}\mathbf{octagon}\mathbf{}\mathbf{is}\mathbf{}\mathbf{regular},\mathrm{that}\mathrm{is}\mathrm{all}\mathrm{its}8\mathrm{sides}\mathrm{are}\mathrm{of}\mathrm{equal}\mathrm{measure}\mathrm{and} \\ \mathrm{interior}\mathrm{angles}\mathrm{at}\mathrm{all}\mathrm{the}\mathrm{vertex}\mathrm{are}\mathrm{also}\mathrm{of}\mathrm{same}\mathrm{measure}. \\ \mathrm{Then}\mathrm{in}\mathrm{this}\mathrm{case},\mathrm{it}\mathrm{has}\mathbf{8}\mathbf{}\mathbf{lines}\mathbf{}\mathbf{of}\mathbf{}\mathbf{symmetry},\mathrm{which}\mathrm{are}\mathrm{at}\mathrm{equal}\mathrm{distance}\mathrm{from} \\ \mathrm{the}\mathrm{center}\mathrm{of}\mathrm{octagon}. \\ \mathrm{Out}\mathrm{of}\mathrm{these},4\mathrm{lines}\mathrm{of}\mathrm{symmetry}\mathrm{of}\mathrm{the}\mathrm{regular}\mathrm{octagon}\mathrm{passes}\mathrm{through}\mathrm{the}\mathrm{vertex} \\ \mathrm{of}\mathrm{the}\mathrm{octagon}\mathrm{and}\mathrm{meet}\mathrm{the}\mathrm{opposite}\mathrm{vertex}\mathrm{of}\mathrm{octagon}. \\ \mathrm{The}\mathrm{remaining}4\mathrm{lines}\mathrm{of}\mathrm{symmetry}\mathrm{passes}\mathrm{through}\mathrm{the}\mathrm{mid}\mathrm{points}\mathrm{of}\mathrm{the}\mathrm{edges} \\ \mathrm{of}\mathrm{octagon}. \\ \mathbf{The}\mathbf{}\mathbf{same}\mathbf{}\mathbf{result}\mathbf{}\mathbf{is}\mathbf{}\mathbf{true}\mathbf{}\mathbf{for}\mathbf{}\mathbf{regular}\mathbf{}\mathbf{Nonagon}\mathbf{}\mathbf{and}\mathbf{}\mathbf{regular}\mathbf{}\mathbf{Decagon} \end{gathered}$$