Question

State whether a given statement is true or false

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

• A. True
• B. False

Answer 1

The correct option is A True

$\because AO=BD=CO=DO$

$\therefore$ In $△AOD$

$\angle OAD+\angle ODA={180}^0-{90}^0={90}^0$

$\because \angle OAD=\angle ODA$ $(\because AO=OD)$

$\because \angle OAD={45}^0$

Similarly, $\angle OAB={45}^0$

$\therefore \angle BAD=\angle DAD+\angle DAB$

$={90}^0$

In $△AOD$ and $△COD$

$AO=OC$

$OD$ is common

$\angle AOD=\angle DOC={90}^0$

$\therefore △AOD\cong △COD$

$\therefore AD=DC$

Similarly

$CD=BC=AB=DA$

$\therefore ABCD$ is a square