Question

In the figure, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD.

Answer 1

Construct a line passing through AD

Now, AD and CD are tangents to the circle with centre O from the external point D.

So, AD = CD   (Tangents drawn from an external point to a circle are equal)   ...(1)

Also, AB and AD are the tangents to the circle with centre O' from the external point A.

So, AD = AB   (Tangents drawn from an external point to a circle are equal)  ...(2)

From (1) and (2)

AB = CD

Hence proved. 

Answer 2

Given: AB and CD are two common tangents to two circles of unequal radii.

To Prove: AB = CD

Construction: Produce AB and CD, to intersect at P.

Proof: Consider the circle with greater radius.

AP = CP  ...[Tangents drawn from an external point to a circle are equal] [1]

Also, Consider the circle with smaller radius.

BP = BD   ...[Tangents drawn from an external point to a circle are equal] [2]

Substract [2] from [1], we get

AP – BP = CP – BD

AB = CD

Hence proved.