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