Question

The diagonals of a quadrilateral bisects each other at right angles. Show that the quadrilateral is a rhombus.

Answer 1

In the given quadrilateral ABCD diagonals AC and BD bisect each other at right angle.

In $\Delta AOB$ and $\Delta AOD$

$DO=OB$ (O is mid point)

$AO=AO$ (Common side)

$\angle AOD=\angle AOB$ (Right angle given )

$\therefore \Delta AOB\cong \Delta AOD$

So, $AB=AD$

In $\Delta BOC$ and $\Delta DOC$

$DO=OB$ (O is mid point)

$OC=OC$ (Common side)

$\angle BOC=\angle DOC$ (Right angle given )

$\therefore \Delta BOC\cong \Delta DOC$

So $BC=CD$

Then $AB=BC=CD=AD$

Then, $ABCD$ is a rhombus.