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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. 10.25).

Q: 4    If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that \small AB=CD (see Fig. \small 10.25).
 

            

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Given: a line intersects two concentric circles (circles with the same centre) with centre O at A, B,  C and D.

To prove: AB  = CD

Construction: Draw OM\perp AD

Proof : 

BC is a chord of the inner circle and OM\perp BC

So, BM = CM  .................1

(Perpendicular OM bisect BC)

Similarly,

AD  is a chord of the outer circle and OM\perp AD

So, AM = DM  .................2

(Perpendicular OM bisect AD )

Subtracting   1 from  2, we get

 AM-BM = DM - CM

\Rightarrow AB = CD

 

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seema garhwal

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