Question

In a rhombus ABCD , O is any interior point such that OA=OC , then prove that D, O and B are collinear.

Answer 1

D, O, and B are collinear .

GIVEN

ABCD is a rhombus.

O is any interior point such that OA=OC.

TO FIND

D, O and B are collinear.

SOLUTION

We can simply solve the above problem as follows;

In ΔCOB and ΔAOB

OA = OC (given)

AB = BC (Side of a rhombus)

OB = OB (Common side)

By S-S-S congruency;

ΔAOB ≈ ΔCOB

Therefore,

∠AOB = ∠COB (i)

Similarly,

ΔAOD ≈ ΔCOD

∠AOB = ∠BOC (ii)

∠AOB + ∠BOC + ∠COD + ∠AOD = 360°

From (i) and (ii)

∠BOC + ∠BOC + ∠COD + ∠COD = 360°

2∠BOC + 2∠COD = 360°

∠BOC + ∠COD = 360/2 = 180°

∠BOC + ∠COD = ∠DOB

Therefore,

∠DOB = 180°

Therefore, D, O, and B are collinear .

*Picture for refernce.

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