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, respectively.

So, OA = OB = OC = OD = r   ...(i)

∵ `bar(AXB) ≅ bar(CYD)`

∴ ∠AOB = ∠COD   ...(ii) [Congruent arcs of a circle subtend equal angles at the centre]

In ΔAOB and ΔCOD,

AO = CO   ...[From (i)]

BO = DO  ...[From (i)]

∠AOB = ∠COD   ...[From (ii)]

∴ ΔAOB ≅ ΔCOD   ...[By SAS congruency]

`\implies` AB = CD   ...[By C.P.C.T.]

`\implies (AB)/(CD) = 1`

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