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.

Steps of construction:

Draw AB = 8 cm. With A and B as centres, radii as 4 cm and 3 cm respectively draw two circles.

Draw the perpendicular bisector of AB, intersecting AB at O.

With O as the centre and OA as radius draw a circle that intersects the two circles at P, Q, R and S.

Join BP, BQ, AR and AS.

BP, BQ are the tangents from B to the circle with centre A.

AR, AS are the tangents from A to the circle with centre B.

Proof:

∠APB = ∠AQB = 90° (Angle in a semi-circle)

∴ AP ⊥ PB and AQ ⊥ QB

Therefore, BP and BQ are the tangents to the circle with centre A.

Similarly, AR and AS are the tangents to the circle with centre B.