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If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig.)

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Given : A line AD intersects two concentric circles at A, B, C and D, where O is the centre of these circles. 

To prove : AB = CD 

Construction : Draw OM ⊥ AD. 

Proof : AD is the chord of larger circle. 

∴ AM = DM ..(i) [OM bisects the chord] 

BC is the chord of smaller circle 

∴ BM = CM ..(ii) [OM bisects the chord] 

Subtracting (ii) from (i), we get 

AM – BM = DM – CM 

⇒ AB = CD Proved.

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