Question

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

Answer 1

The diagonals are $AC$ and $BD$

Let the diagonals bisect each other at $O$.

In $\Delta AOBand\Delta AOD$

$OA=OA$ (common)

$OB=OD$ (given the bisect)

$\angle AOB=\angle AOD$ (each ${90}^0$)

$\therefore \Delta AOB\cong \Delta AOD$ (SAS criteria)

The corresponding parts are equal.

$AB=AD$

Similarly, $AB=BC$

$BC=CD$

$CD=AD$

$\therefore AB=BC=CD=DA$

i.e. the quadrilateral is a Rhombus