Question

Two alternate sides of a regular polygon, when produced, meet at right angles. Find the number of sides of the polygon.

• A. 6
• B. 8
• C. 10
• D. 20

Answer 1

The correct option is B

8

Let the number of sides of the regular polygon = n

AB and DC when produced meet at P such that

$\angle P={90}^{\circ }$

(because Interior angles of a regular polygon are equal)

$\Rightarrow \angle ABC=\angle BCD$

$\Rightarrow {180}^{\circ }-\angle ABC={180}^{\circ }-\angle BCD$

$\Rightarrow \angle PBC=\angle PCB$

Given $\angle P={90}^{\circ }$

$\therefore \angle PBC+\angle PCB={180}^{\circ }-{90}^{\circ }={90}^{\circ }$

$\therefore \angle PBC=\angle PCB=\frac{1}{2}\times {90}^{\circ }={45}^{\circ }$

Each exterior angle

$={45}^{\circ }$

$\Rightarrow {45}^{\circ }=\frac{{360}^{\circ }}{n}$

$\Rightarrow n=\frac{{360}^{\circ }}{{45}^{\circ }}$

$\Rightarrow n=8$