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.

Answer 1

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.