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

Construction : Draw $OP\perp AD$


For outer circle, $AD$ is chord
$\therefore AP=PD$ ...(1) ($\because \perp$ from centre bisect the chord)
& for inner circle, $BC$ is chord
$\therefore BP=PC$ ....(2) ($\because \perp$ from centre bisect the chord)
Subtract equation (2) from equation (1)
$\Rightarrow AP-BP=PD-PC$
$\Rightarrow AB=CD$