Two alternate sides of a regular polygon, when produced, meet at right angle. Find :

(i) the value of each exterior angle of the polygon;

(ii) the number of sides in the polygon.

(i) Let AB and CD be the alternate sides of regular polygon.

Given,

Two alternate sides of a regular polygon, when produced, meet at right angle.

We know that,

Interior angles of regular polygon are equal.

∴ ∠ABC = ∠BCD

⇒ 180° - ∠ABC = 180° - ∠BCD

⇒ ∠PBC = ∠BCP = x

In △ PBC,

⇒ ∠PBC + ∠BCP + ∠BPC = 180°

⇒ x + x + 90° = 180°

⇒ 2x = 180° - 90°

⇒ 2x = 90°

⇒ x = $\dfrac{90°}{2}$

⇒ x = 45°.

∴ ∠PBC = ∠BCP = 45°.

Hence, value of each exterior angle of the polygon = 45°.

(ii) By formula,

Number of sides in polygon = $\dfrac{360°}{\text{Exterior angle}} = \dfrac{360°}{45°}$ = 8.

Hence, number of sides in the polygon = 8.