Question

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

Answer 1

Given,

•Radius of the circle is 4 cm.

•Angle between the tangents is 60°.

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. Extend the line segment OA to B such that OA=AB=4 cm.

3. Taking A as the centre draw a circle of radius AO=AB=4 cm. This circle intersects the first 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

So, ∠PAO=60°

Now,

∠BAP + ∠PAO = 180° (linear angle)

∠BAP + 60° = 180°

∠ BAP = 60°

In ∆BAP, we have

BA=AP = 4 cm (radii of the circle with centre A)

∠BAP=120°

As two sides BA and AP are equal rABP is isosceles.

So, ∠ABP=∠APB

Let ∠ABP=∠APB = α

As the sum of angles is a triangle is 180°

∠ABP + ∠APB + ∠BAP = 180°

α + α + 120° = 180°

2α = 60°

α = 30°

Therefore ∠ABP=∠APB=30°

Hence ∠PBQ=60°