To Prove: If diagonals of a quadrilateral bisect at 90º, it is a rhombus.
Definition of Rhombus: A parallelogram whose all sides are equal.
Given: Let ABCD be a quadrilateral whose diagonals bisect at 90º
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)
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 rhombusHence, Proved.
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