Question

If a line intersects two concentric circles (circles with the same centre) with centre $\mathrm{O}$ at $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$, prove that $\mathrm{AB}=\mathrm{CD}$ (see Fig) .

Answer 1

Solve for the required proof:

Draw a perpendicular $\mathrm{OM}$ from $\mathrm{O}$.

As perpendicular from center of circle to the chord bisects the chord.

So, in inner circle, $\mathrm{BM}=\mathrm{MC}$ $...\left(\mathrm{i}\right)$

And for outer circle, $\mathrm{AM}=\mathrm{MD}$ $...\left(\mathrm{ii}\right)$

Now subtract $\left(\mathrm{i}\right)$ from $\left(\mathrm{ii}\right)$, we get;

$\mathrm{AM}-\mathrm{BM}=\mathrm{MD}-\mathrm{MC}$

$\Rightarrow$ $\mathrm{AB}=\mathrm{CD}$

Hence proved.