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 ......(i)
arc AXB $\cong$ arc CYD

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

$In\Delta AOBand\Delta COD$
AO = CO
BO = DO
$\angle AOB=\angle COD$
$\therefore \Delta AOB\cong \Delta COD$ [by SAS congruence rule]
$\Rightarrow$ AB = CD [by CPCT)
$\Rightarrow \frac{AB}{CD}=1$
Hence, the ratio of AB and CD is 1:1