Question

If a line l intersects two concentric circles at points A, B, C, and D as shown in the figure, prove that AB = CD. 

OR 

Prove that two concentric circles intercept equal portions on any straight line that cuts them.

Answer 1

Let O be the centre of the two concentric circles and OM the perpendicular from O to the line l. AD is the chord of the larger circle and BC, the chord of the smaller circle.

Since, perpendicular from the centre to a chord bisects the chord, therefore, 

AM = MD ...(i) 

and BM = MC ...(ii)

 (i) and (ii) gives, AM – BM = MD – MC 

⇒ AB = CD.