Question

Prove that the quadrilateral formed by joining the mid point of the pair of adjacent sides of rectangle is a rhombus

Answer 1

Given AC,BD are diagonals of a quadrilateral ABCD are perpendicular. P,Q,R and S are the mid points of AB,BC, CD and AD respectively.

Proof: In  ΔABC, P and Q are mid points of AB and BC respectively. PQ|| AC

PQ =½AC ..................(1) (Mid point)
Similarly in ΔACD, R and S are mid points of sides CD and AD respectively.
  SR||AC and SR=½AC ...............(2) (Mid point theorem)
 From (1) and (2), we get PQ||SR and PQ =SR

 Hence, PQRS is parallelogram ( pair of opposite sides is parallel and equal)

Now, RS || AC and QR || BD.

Also, AC ⊥ BD (Given)

∴RS ⊥ QR. 

Thus, PQRS is a rectangle.