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