Question

Two alternate sides of a regular polygon, when produced, meet at the right angle, Calculate the number of sides in the polygon.

Answer 1

Let number of sides of regular polygon = n

AB & DC when produced meet at P such that $\angle P={90}^o$

$\because$ Interior angles are equal.

$\therefore \angle ABC=\angle BCD\therefore {180}^o-\angle ABC={180}^o-\angle BCD\therefore \angle PBC=\angle BCPBut\angle P={90}^o\left(Given\right)\therefore \angle PBC+\angle BCP={180}^o-{90}^o={90}^o\therefore \angle PBC=\angle BCP=\frac{1}{2}\times {90}^o={45}^o$

$\therefore$ Each exterior angle $={45}^o$

$\therefore {45}^o=\frac{{360}^o}{n}n=\frac{{360}^o}{{45}^o}n=8$