Draw line segment AB of length …

 
To construct: A line segment of length 8 cm and taking A as centre, to draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Also, to construct tangents to each circle from the centre to the other circle.
Steps of Construction :

1. Bisect BA. Let M be the mid-point of BA.
2. Taking M as centre and MA as radius, draw a circle. Let it intersects the given circle at the points P and Q.
3. Join BP and BQ.
   Then, BP and BQ are the required two tangents from B to the circle with centre A.
4. Again, Let M be the mid-point of AB.
5. Taking M as centre and MB as radius, draw a circle. Let it intersects the given circle at the points R and S.
6. Join AR and AS.
   Then, AR and AS are the required two tangents from A to the circle with centre B.
   Justification: Join BP and BQ.
   Then {tex}\angle APB{/tex} being an angle in the semicircle is 90^o.
   {tex} \Rightarrow BP \bot AP{/tex}
   Since AP is a radius of the circle with centre A, BP has to be a tangent to a circle with centre A. Similarly, BQ is also a tangent to the circle with centre A.
   Again join AR and AS.
   Then {tex}\angle ARB{/tex} being an angle in the semicircle is 90^o.
   {tex} \Rightarrow AR \bot BR{/tex}
   Since BR is a radius of the circle with centre B, AR has to be a tangent to a circle with centre B. Similarly, AS is also a tangent to the circle with centre B.