Prove using vectors: The quadrilateral obtained by joining mid points of adjacent sides of a rectangle is a rhombus.

Explanation

A rectangle `ABCD` in which `P,Q,R and S` are the mid-points of sides `AB,BC,CD and DA` respectively.


`PQ,QR,RS and SP` are joined.


Join `AC`. In `triangleABC`,
`P and Q` are the mid-points of sides `AB` and `BC` respectively.
`therefore PQ|\|AC and PQ= 1/2AC .....(i)` (Mid pt. Theorem)


In `triangleADC `,
`R` and `S` are the mid-points of sides `CD` and `AD` respectively.
`therefore SR|\|AC and SR= 1/2 AC ....(ii) ` (Mid. pt. Theorem)
From `(i) and (ii)`, we have

`PQ|\|SR and PQ=SR`

...