Question

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

Answer 1

We know that, $OA=OD$ and $OB=OC$.

They are radius of respective circles.

In $\Delta OBC$, we know that $OB=OC$, so $\angle OBC=\angle OCB$

$\therefore \angle OCD=\angle OBA$

In $\Delta OAD$, we know that $OA=OD$, so $\angle OAD=\angle ODA$

Since,$\angle OCD=\angle OBA$ and $\angle OAD=\angle ODA$, we get $\angle AOB$ in $\Delta OAB$ is equal to $\angle COD$ in $\Delta OCD$.

$\therefore$ From $SAS$ congruency, we can say that $\Delta OAB$ and $\Delta OCD$ are congruent.

So, $AB=CD$.