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Question
A point O is taken inside a rhombus ABCD such that its distance from the vertices B and D are equal. Show that AOC is a straight line.
Solution
In ΔAOD and ΔAOB,
AD = AB ...(given)
AO = AO ...(Common)
OD = OB ...(given)
⇒ ΔAOD ≅ ΔAOB ...(by SSS congruence criterion)
⇒ ∠AOD = ∠AOB ...(c.p.c.t.) ...(i)
Similarly, ΔDOC ≅ ΔBOC
⇒ ∠DOC = ∠BOC ...(c.p.c.t.) ...(ii)
But, ∠AOB + ∠AOD + ∠COD + ∠BOC = 4 Right angles ...[ Sum of the angles at a point is 4 Right angles ]
⇒ 2∠AOD + 2∠COD = 4 Right angles ....[ Using (i) and (ii) ]
⇒ ∠AOD + ∠COD = 2 Right angles
⇒ ∠AOD + ∠COD = 180°
⇒ ∠AOD and ∠COD form a linear pair.
⇒ AO and OC are in the same straight line.
⇒ AOC is a straight line.
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