Question

2.Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.​

Answer 1

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Given that,

OA = OC

OB = OD

and ∠AOB = ∠BOC = ∠OCD = ∠ODA = 90°

AC = BD

To Prove

If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Solution

Let ABCD be a quadrilateral whose diagonals bisect each other at right angles.

We have to prove that ABCD is parallelogram and AB = BC = CD = AD

From ΔAOB and ΔCOB,

OA = OC (Given)

∠AOB = ∠COB (Opposite sides of a parallelogram are equal)

OB = OB (Common)

Therefore, ΔAOB ≅ ΔCOB [SAS congruency]

Thus, AB = BC [CPCT]

Similarly we can prove that

BC = CD

CD = AD

AD = AB

AB = BC = CD = AD

Opposites sides of a quadrilateral are equal hence ABCD is a parallelogram.

ABCD is rhombus as it is a parallelogram whose diagonals intersect at right angle.

Thus, Proved.