Question

Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60º. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.

Answer 1

In order to draw the pair of tangents, we follow the following steps:

Steps of construction:

1. Take a point O on the plane of the paper and draw a circle of radius OA = 4 cm.
2. Produce OA to B such that OA = AB = 4 cm.
3. Taking A as the centre draw a circle of radius AO = AB = 4 cm. Suppose it cuts the circle drawn in step 1 at P and Q.
4. Join BP and BQ to get desired tangents.

Justification:

In ∆OAP, we have

OA = OP = 4 cm  ...(∵ Radius)

Also, AP = 4 cm  ...(∵ Radius of circle with centre A)

∴ ∆OAP is equilateral

⇒ ∠PAO = 60°

⇒ ∠BAP = 120°

In ∆BAP, we have

BA = AP and ∠BAP = 120°

∴ ∠ABP = ∠APB = 30°

⇒ ∠PBQ = 60°