Question

The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is rectangle.

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. Diagonal AC and BD intersect at a right angle at point O. We need to show that PQRS is a rectangle

Proof:

From and ΔABC and ΔADC
2PQ = AC and PQ || AC              …..(1)
2RS = AC and RS || AC               …..(2)

From (1) and (2) we get,
PQ = RS and PQ || RS
Similarly, we can show that PS=RQ and PS || RQ

Therefore PQRS is a parallelogram.
Now PQ || AC, therefore  ∠AOD = ∠PXO = 90°      ...[ Corresponding angel ]

Again BD || RQ, therefore ∠PXO = ∠RQX = 90°  ...[ Corresponding angel]

Similarly ∠QRS = ∠RSP = ∠SPQ = 90°  
Therefore PQRS is a rectangle.
Hence proved.