Question

Prove that if the diagonals of a quadrilateral are equal and are perpendicular bisectors to each other, then it is a square.[4 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Proof : 2 Marks


Let $ABCD$ be a quadrilateral whose diagonal $AC$ and $BD$ are equal and are perpendicular bisector of each other.

Thus, $\angle AOB=\angle BOC=\angle COD=\angle DOA=90°$

and $AO=BO=CO=DO.$

Consider triangles $\Delta AOB$ and $\Delta BOC$, COD and DOA

$AO=CO$ [Given]

$BO=BO$ [Common]

$\angle AOB=\angle BOC$ [Given]

$\Rightarrow \Delta AOB\cong \Delta COB$ [SAS congruency rule]

$AB=BC$ [CPCTC]

Similarly

$\Delta COB\cong \Delta COD\Rightarrow CB=CD$

$\Delta COD\cong \Delta AOD\Rightarrow CD=AD$

$\Delta AOD\cong \Delta AOB\Rightarrow AD=AB$
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$\therefore AB=BC=CD=DA$ -------(i)

Moreover, since all four are isosceles right triangles.

$\angle DAO=\angle BAO={45}^{\circ }$

Or, $\angle DAB={90}^{\circ }$ -----(ii)

Combining (i) and (ii), we can say that the quadrilateral ABCD is a square.