Question

If arcs $AXB$ and $CYD$ of a circle are congruent, find the ratio of $AB$ and $CD.$

Answer 1

Let AXB and CYD are arcs of a circle whose centre and radius are O and r units, respectively.
So, $OA=OB=OC=OD=r......\left(i\right)$
arc $AXB\cong$ arc $CYD$

$\angle AOB=\angle COD$ [congruent arcs of a circle subtend equal angles at the centre]$.....\left(ii\right)$

In $\Delta AOB$ and $\Delta COD$, we have

$AO=CO$ [Radii]

$BO=DO$ [Radii]

$\angle AOB=\angle COD$ [from above $eq.\left(ii\right)$]

$\therefore \Delta AOB\cong \Delta COD$ [by SAS congruence rule]

$\Rightarrow AB=CD$ [by CPCT)

$\Rightarrow \frac{AB}{CD}=1$

Hence, the ratio of $AB$ and $CD$ is $1:1$