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

Let ABCD be a quadrilateral, whose diagonals AC and BD bisect each other at right

Angle i.e., OA = OC, OB = OD, and

∠AOB =∠ BOC = ∠COD = ∠AOD = 90⁰. To prove

ABCD a rhombus, we have to prove ABCD is a parallelogram and all the sides of ABCD are equal.

In ΔAOD and ΔCOD,

OA = OC (Diagonals bisect each other)

∠AOD = ∠COD (Given)

OD = OD (Common)

ΔAOD ΔCOD (By SAS congruence rule)

AD = CD (1)

Similarly,

AD = AB and CD = BC (2)

From equations (1) and (2),

AB = BC = CD = AD

Since opposite sides of quadrilateral ABCD are equal, it can be said that ABCD is a parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that

ABCD is a rhombus