If the diagonals of a quadrilateral ABCD bisect each other at the point O then the quadrilateral ABCD can be a _____.

Trapezium

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Kite

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Parallelogram

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None of the above

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Solution

The correct option is C Parallelogram
From the given figure, consider the triangles AOB and COD.
AO = OC (Given)
BO = OD (Given)
$∠ A O B = ∠ C O D (v e r t i c a l l y o p p o s i t e a n g l e s) △ A O B ≅ △ C O D$

By corresponding parts of congruent triangles,
$∠ A B O = ∠ C D O$ -------(i)

But, (i) forms alternate interior angles. Hence, AB || CD.
Similarly, AD || BC.
Since, both the pair of opposite sides are parallel, quadrilateral ABCD is a parallelogram.

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