Question

In the figure, if a line intercepts two concentric circles with centre $O$ at $A,B,C$ and $D$, prove that $AB=CD$.

Answer 1

Here, $\Delta AOD$ $\Rightarrow$

$AO=DO$[radius]

and, OM intersects AD

equally $[\because OM\perp AD]$

So, $AM=MD$ ……….$\left(1\right)$

Again, $\Delta BOC\Rightarrow$

$BO=CO$ [radius]

and, OM also intersects BC equally$[\because OM\perp BC]$

So, $BM=MC$ ……$\left(2\right)$

$\left(1\right)\Rightarrow$

$AM=MD$

$\Rightarrow AB+BM=MC+CD$

$\Rightarrow AB+BM=BM+CD$ $\left[\right(2\left)\right]$

$\Rightarrow AB=CD$ [Proved].