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


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