Question

Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Answer 1

Let $\mathrm{ABCD}$ be a quadrilateral where its diagonals $\mathrm{AC}$$=$ $\mathrm{BD}$ bisect each other at right angle at $\mathrm{O}$ i.e. $\mathrm{AO}=\mathrm{OC}$ and $\mathrm{BO}=\mathrm{OD}$

To prove:

The quadrilateral $\mathrm{ABCD}$ is a square.

In $△\mathrm{AOB}$ and $△\mathrm{COD}$, we have

$\mathrm{AO}=\mathrm{OC}$ (given)

$\angle \mathrm{AOB}=\angle \mathrm{COD}$ (vertically opposite angles)s

$\mathrm{BO}=\mathrm{OD}$ (given)

⇒ $△\mathrm{AOB}$$\cong$$△\mathrm{COD}$ by $\mathrm{SAS}$(side- angle- side) criteria

So,$\angle \mathrm{ABO}=\angle \mathrm{CDO}$ and $\mathrm{AB}=\mathrm{CD}$ by $\mathrm{CPCT}$(corresponding parts of congruent triangle)

If pair of opposite sides of a quadrilateral are equal and parallel then it is parallelogram. In quadrilateral $\mathrm{ABCD}$ , $\angle \mathrm{ABO}=\angle \mathrm{CDO}$(alternate interior angles) which implies that $\mathrm{AB}\left|\right|\mathrm{CD}$ and $\mathrm{AB}=\mathrm{CD}$. So, $\mathrm{ABCD}$ is a parallelogram in which $\mathrm{AB}=\mathrm{CD}$ and $\mathrm{AD}=\mathrm{BC}$ (opposite sides of parallelogram are equal)

In $△\mathrm{AOD}$ and $△\mathrm{COD}$, we have

$\mathrm{AO}=\mathrm{OC}$ (given)

$\angle \mathrm{AOD}=\angle \mathrm{COD}={90}^0$ (given)

$\mathrm{OD}=\mathrm{OD}$ (common)

⇒ $△\mathrm{AOD}$$\cong$$△\mathrm{COD}$ by $\mathrm{SAS}$(side- angle- side) criteria

So, and $\mathrm{AB}=\mathrm{AD}$ by $\mathrm{CPCT}$(corresponding parts of congruent triangle)

Similarly, $△\mathrm{BOC}$$\cong$$△\mathrm{DOC}$ by $\mathrm{SAS}$(side- angle- side) criteria

So, and $\mathrm{BC}=\mathrm{CD}$ by $\mathrm{CPCT}$(corresponding parts of congruent triangle)

: Based on the above three steps it is evident that $\mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DA}$

Here, all the sides of the parallelogram $\mathrm{ABCD}$ are equal and its diagonals bisect each other at right angle .Thus, $\mathrm{ABCD}$ is a square.