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.