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