Consider △ ABC
We know that P and Q are the midpoints of AB and BC
Based on the midpoint theorem
We know that PQ || AC and PQ = ½ AC
Consider △ ADC
Based on the midpoint theorem
We know that RS || AC and RS = ½ AC
It can be written as PQ || RS and
PQ = RS = ½ AC ……. (1)
Consider △ BAD
We know that P and S are the midpoints of AB and AD
Based on the midpoint theorem
We know that PS || BD and PS = ½ DB
Consider △ BCD
We know that RQ || BD and RQ = ½ DB
It can be written as PS || RQ and
PS = RQ = ½ DB ……… (2)
We know that the diagonals of a rectangle are equal
It can be written as
AC = BD ………. (3)
Comparing equations (1), (2) and (3)
We know that
PQ || RS and PS || RQ
So we get
PQ = QR = RS = SP
Therefore, it is proved that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rectangle is a rhombus.
