Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

Construction Procedure:

The given circle’s tangents can be constructed as follows.

1. Construct a line segment AB of measure 8 cm.

2. Draw a circle of radius 4 cm taking A as centre.

3. Draw a circle of radius 3 cm taking B as centre,

4. Taking the midpoint as M draw a perpendicular bisector of the line AB.

5. Now, draw a circle with the radius of MA or MB by taking M as centre which the intersects the circle at the points P, Q, R and S.

6. Join AR, AS, BP and BQ now.

7. As a result, AR, AS, BP and BQ are the required tangents.

Justification:

The construction can be justified by demonstrating that AS and AR are tangents to the circle (whose radius is 3 cm and centre is B) and BP and BQ are tangents to the circle (whose radius is 4 cm and centre is A).

To prove this, now join AP, AQ, BS, and BR.

∠ASB denotes as an angle in the semi-circle. It is known that an angle in a semi-circle is a right angle.

Therefore, ∠ASB = 90°

⇒ BS ⊥ AS

AS must be a tangent of the circle, since BS is the radius of the circle.

Similarly, the required tangents of the given circle are AR, BP, and BQ.