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