Question

The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.

Answer 1

To prove: ABCD is a square,
that is, to prove that sides of the quadrilateral are equal
and each angle of the quadrilateral is 90°,
ABCD is a rectangle,
⇒ ∠A = ∠B = ∠c = ∠D = 90° and diagonals bisect each other.

that is, MD = BM                     ....(i)
Consider ΔAMD and ΔAMB,
MD =  BM                               ....( from(i) )
∠AMD = ∠AMB = 90°             .....(given)
AM = AM                                ......( common side )
ΔAMD ≅ ΔAMB                      ....(SAS ngruence iterion)
⇒ AD = AB                              ....( c.p.c.t.c. )
Since ABCD is a rectangle, AD = BC and AB = CD
Thus, AB = BC = CD = AD and ∠A = ∠B = ∠C = ∠D = 90°
⇒ ABCD is a square.