`` ABCD is a rhombus and `P ,Q ,R ,S` are the mid-points of `A B ,B C ,C D ,D A` respectively. Prove that ` P Q R S` is a rectangle.

Explanation

Given: `ABCD` is a rhombus and `P`, `Q`, `R` and `S` are the mid-points of the sides `AB`, `BC`, `CD` and `DA` respectively.
To Prove: `PQRS` is a rectangle.
Construction: Diagonals `AC` and `BD` are drawn.
Proof: In `/_\ADC`,
`S` and `R` are the mid points of `AD` and `DC`
`:. SR` parallel to `AC`
`SR=1/2AC` (By mid-point formula)
In `/_\ABC`,
`P` and `Q` are the mid-points `AB` and `BC`
`:.PQ` parallel to `AC`
`PQ=1/2AC`
`∴ SR` parallel `PQ` and `SR=PQ`
`:.PQRS` is a parallelogram.
But diagonals of a rhombus bisect at right angles.
`90^@` angle is formed at `O`
`:./_P=90^@`
`:.PQRS` is a parallelogram, each of its angle is right angle.
This is the property of rectangle.
`:.PQRS` is a rectangle.
Hence Proved.