Question

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

• A. True
• B. False

Answer 1

The correct option is A True

Given: Quadrilateral ABCD. AC and BD intersect at O such that they bisect each other at right angles. AC = BD.
Since, the diagonals bisect each other and they are equal,
$OA=OB=OC=OD=x$
In $△OAB$
$A{B}^2=O{A}^2+O{B}^2$ (Pythagoras theorem)
$A{B}^2=x^2+x^2$
$AB=x\sqrt{2}$
Similarly, $AD=BC=CD=x\sqrt{2}$
Hence, all the sides are equal.
Now, In $△OAB$
since, $OA=OB$
hence, $\angle OAB=\angle OBA=y$
Sum of angles of triangle OAB = 180
$\angle OAB+\angle OBA+\angle AOB=180$
$y+y+90=180$
$2y=90$
$y={45}^{\circ }$
$\angle OAB=\angle OBA={45}^{\circ }$
Similarly, $\angle OBC=\angle OCB={45}^{\circ }$
Thus, $\angle B=\angle OBA+\angle OBC=45+45={90}^{\circ }$
Similarly, $\angle A=\angle C=\angle D={90}^{\circ }$
Hence, all the sides are equal and adjacent sides are perpendicular to each other. This quadrilateral is a square.