Question

Suppose in a rectangle, the diagonals are perpendicular to each other. Prove that it is a square.

Answer 1

Consider two $△AED$ $\xi$ $△AEB$

Now, $AE=AE$ [ Because they are the same line ]

$\Rightarrow ED=EB$ [ Diagonals of rectangle bisect each other ]

and $\angle AED=\angle AEB.$ (Given)

$\therefore △AED\cong △AEB$ [ by SAS ]

Then we have $AD=AB$........(1)

Again in this rectangle $AD=BC$.........(2) and $CD=AB$......(3).

From (1) , (2) and (3) we have $AB=BC=CD=DA$ with the angles right-angle.

So, ABCD is a square.