Question

If AB and CD are the common tangents in the circles of two unequal (different) radii, then show that seg AB ≅ seg CD.

Answer 1

Given: AB and CD are tangents to both circles.

To prove: seg AB ≅ seg CD

Construction: Extend seg AB and seg CD to intersect each other at point E, such that A – B – E, C – D – E.

Proof: In above figure,

`{:(AE = CE),(BE = DE):}}`   ...(i) [Tangent segment theorem]

Consider, AE = CE

∴ AB + BE = CD + DE   ...[A – B – E, C – D – E]

∴ AB + DE = CD + DE   ...[From (i)]

∴ AB = CD

∴ seg AB ≅ seg CD