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

Given: ABCD is a quadrilateral in which diagonals AC and BD are equal and bisect each other at right angles.

To prove: ABCD is a square.

Proof: ABCD is a quadrilateral whose diagonals bisect  each other. 

So it is a parallelogram. 

Also its diagonals bisect each other at right angles.

So ABCD is a rhombus.

⇒ AB = BC = CD = DA

In ΔABC and ΔABD, we have

AB = AB (common sides)

BC = AD (side of a rhombus are equal)

AC = BD (given)

ΔABC = ΔBAD (by SSS congruency rule)

∠ABC = ∠BAD (by c.p.c.t)

But ∠ABC + ∠BAD = 180° (sum of the consecutive interior angles)

⇒ ∠ABC = ∠BAD = 90°

⇒ ∠A = ∠B = ∠C = ∠D = 90° (opposite angles of a parallelogram)

⇒ ABCD is a rhombus whose angles are of 90° each.

Hence, ABCD is a square.