1.

Show that if the diagonal of a quadrilateral bisect each other at right angle,then it is a rhombus​

Answer»

Answer:-

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

★ GIVEN that:-

OA = OC

OB = OD

∠AOB = ∠BOC = ∠OCD = ∠ODA = 90°

★ To show that:-

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

★ To PROVE that:-

ABCD is parallelogram and AB = BC = CD = AD

★ Proof,

In 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,

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.

Hence Proved.


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