Question

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.

Answer 1

Data: A line intersects two concentric circles with centre O at A, B, C and D. 

To Prove: AB = CD. 

Construction: Draw OM⊥AD. 

Proof: AD is the chord of bigger circle. OM⊥AD 

∴ AM = MD ………(i) 

(∵ Perpendicular bisects the chord from centre) 

Similarly, BC is the chord of smaller circle. OM⊥BC. 

∴ BM = MC ………. (ii) 

Subtracting (ii) from (i), 

AM – BM = MD – MC 

∴ AB = CD.