Question

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

Answer 1

P and Q are mid-points of AB and BC.

∴ PQ || AC and PQ = `(1)/(2)"AC"`.......(i)

S and R are mid-points of AD and DC.

∴ SR || AC and SR = `(1)/(2)"AC"`.......(ii)
From (i) and (ii)
PQ || SR and PQ = SR
Therefore, PQRS is a parallelogram.
Further AC and BD intersect at right angles
∴ SP  || BD and BD ⊥ AC
∴ SP ⊥ AC
⇒ SP ⊥ SR
⇒ ∠RSP = 90°
∴ ∠ RSP = ∠SRQ = ∠RQS = ∠SPQ = 90°
Therefore, PQRS is a rectangle.