Question

Prove that the rhombus with equal diagonals is a square.

Answer 1

$ABCD$ is a rhombus, in which diagonals AC and BD are equal.

We know that diagonals of rhombus bisect each other.

As $AC=BD$

$\therefore AO=BO=CO=DO$

In $△AOB,$

$\Rightarrow AO=OB$ and $\angle AOD=90°$

$\therefore \angle OAB=\angle OBA=\frac{{90}^o}{2}=45°$

Similarly in $△AOD,$

$\Rightarrow \angle OAD=\angle ODA=45°$

$\therefore \angle A=\angle OAB+\angle OAD={45}^o+{45}^o=90°$

Similarly,

$\Rightarrow \angle B=\angle C=\angle D=90°$

here, $AB=BC=CD=DA$

$\therefore$ Quadrilateral ABCD is a square.