# 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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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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