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

To find: Show that the quadrilateral is square.

Given: The diagonals of a quadrilateral are equal and bisect each other at right angles.

Concept Used:

Diagram:

Explanation:

From the figure we can see that,

OA = OB = OC = OD

In Δ OAB,

∠ OAB = ∠ OBA

And ∠ AOB = 90˚

Sum of angles of a triangle = 180˚

∠ OAB + ∠ OBA + 90˚ = 180˚

2 ∠ OAB = 90˚

∠ OAB = 45˚

In Δ OAD,

∠ OAD = ∠ ODA

And ∠ AOD = 90˚

Sum of angles of a triangle = 180˚

∠ OAD + ∠ ODA + 90˚ = 180˚

2 ∠ OAD = 90˚

∠ OAD = 45˚

Now, we can see that,

∠ OAD + ∠ OAB = 90˚

∠ BAD = 90˚

Therefore, all the angles of the quadrilaterals are 90˚

And we can see that all the four triangles are congruent.

As all sides are equal and all angles are right angles of quadrilateral ABCD, the quadrilateral is a square.

Hence, proved.