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

Draw a perpendicular OM to the line AD.
It can be observed that BC is the chord of the smaller circle and AD is the chord of the bigger circle.
We know that perpendicular drawn from the centre of the circle bisects the chord.
$\therefore BM=MC...\left(1\right)$
And, AM = MD ... (2)
On subtracting equation (2) from (1), we obtain
AM - BM = MD - MC
$\Rightarrow AB=CD$,hence proved.