Question

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

Answer 1

Prove the statement.

Let's draw the figure.

We can see 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 center of the circle bisects the chord.

Thus,

$BM=MC$……………..$\left(1\right)$

$AM=MD$……………...$\left(2\right)$

From equation ,$\left(2\right)-\left(1\right)$

$\Rightarrow AM-BM=DM-CM$

From the figure,

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

Hence, it is proved that $\mathrm{AB}=\mathrm{CD}$.