Question

Draw a circle of radius 4 centimetres and draw a regular pentagon with all its sides touching the circle.

Answer 1

In a polygon, sum of all the sides is given by (n − 2) × 180°, where n is the number of sides.

For a regular pentagon, n = 5

Sum of all the angles = (5 − 2) × 180°

= 3 × 180°

= 540°

Since all the angles of a regular pentagon are equal, measure of each angle = = 108°.

The rough sketch of the required figure can be drawn as follows:

We know that the central angle of the smaller arc between the two points on a circle and the angle between the tangents at these points are supplementary.

∴ ∠PBQ + ∠POQ = 180°

⇒ 108° + ∠POQ = 180°

⇒ ∠POQ = 180° − 108° = 72°

We also know that any tangent to circle is perpendicular to the radius to the point of contact.

∴ ∠OPB = ∠OQB = ∠OTA = ∠OSE = ∠ORD = 90°

We will use these measurements to construct the required figure.

The steps of construction are as follows:

1) Draw a circle with centre O and radius, OQ = 4 cm.

2) Draw ∠OQX of measure 90° at point Q by taking OQ as the base. Extend ray QX upwards.

3) Draw ∠QOP of measure 72° at point O by taking QO as the base such that point P lies on the circle.

4) Draw ∠OPB of measure 90° at point P by taking OP as the base such that B is a point on line QX. Extend ray PB on the left side.

5) Draw ∠POT of measure 72° at point O by taking PO as the base such that point T lies on the circle.

6) Draw ∠OTA of measure 90° at point T by taking OT as the base such that A is a point on line PB. Extend ray TA upwards.

7) Draw ∠TOS of measure 72° at point O by taking TO as the base such that point S lies on the circle.

8) Draw ∠OSE of measure 90° at point S by taking OS as the base such that E is a point on line TA. Extend ray ES upwards.

9) Draw ∠SOR of measure 72° at point O by taking SO as the base such that point R lies on the circle.

10) Draw ∠ORD of measure 90° at point R by taking OR as the base such that D is a point on line SE.

11) Extend ray RD downwards to intersect line QX at point C.

ABCDE is the required pentagon with all the sides touching the circle.