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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.
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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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