Theorem 5: One Pair of Opposite Sides Parallel and Equal
Question 3Sho...
Question
Question 3 Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
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Solution
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=90∘. To prove ABCD a rhombus, we have to prove ABCD is a parallelogram and all the sides of ABCD are equal.
In ΔAODandΔCOD,
OA = OC (Diagonals bisect each other) ∠ AOD =∠ COD (Given)
OD=OD (Common) ∴ΔAOD≅ΔCOD (By SAS congruence rule) ∴ AD=CD ……………(1)
Similarly, it can be proved that
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 AB CD is a rhombus.