Question

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

❌Irrelevant Answer not allowed❌​

Answer 1

Answer:

Let ABCD be a quadrilateral whose diagonals bisect each other at right angles. Therefore, ΔAOB ≅ ΔCOB by SAS congruence condition. Opposites sides of a quadrilateral are equal hence ABCD is a parallelogram. Thus, ABCD is rhombus as it is a parallelogram whose diagonals intersect at right angle.

hope it helps you

Answer 2

Answer:

Take quadrilateral ABCD , AC and BD are diagonals which intersect at O.

In △AOB and △AOD

DO=OB ∣ O is the midpoint

AO=AO ∣ Common side

∠AOB=∠AOD ∣ Right angle

So, △AOB≅△AOD

So, AB=AD

Similarly, AB=BC=CD=AD can be proved which means that ABCD is a rhombus.