Question

If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral ABCD is a rectangle,
show that the diagonals AC and BD intersect at the right angle.

Answer 1

The figure is shown below

Let ABCD be a quadrilateral where P, Q, R, S are the midpoint of AB, BC, CD, DA.PQRS is a rectangle. Diagonal AC and BD intersect at point O. We need to show that AC and BD intersect at a right angle.

Proof:
PQ || AC, therefore ∠AOD = ∠PXO     ...[ Corresponding angle ]...(1)

Again BD || RQ, therefore ∠PXO = ∠RQX = 90°  ....[ Corresponding angle and angle of a rectangle ]...(2)

From (1) and (2) we get ,
∠AOD = 90°

Similarly, ∠AOB = ∠BOC = ∠DOC = 90°
Therefore diagonals AC and BD intersect at right angle.
Hence proved.