Prove that the figure formed by joining the mid-points of the pair of consecutive sides of a quadrilateral is a parallelogram. OR `B A C D` is a parallelogram in which `P ,Q ,R` and `S` are mid-points of the sides `A B ,B C ,C D` and `D A` respectively. `A C` is a diagonal. Show that : `P Q|\| A C` and `P Q=1/2A C` `S R|\| A C` and `S R=1/2A C` `P Q=S R` (iv) `P Q R S` is a parallelogram.

Explanation

Given: In quadrilateral `ABCD`
`P`,`Q`,`R` and `S` are the midpoints of sides `AB`,`BC`,`CD` and `DA`
`AC` is the diagonal
To prove: `PQRS` is a parallelogram
Proof: (i) In `△ABC`,
`P` and `Q` are the mid-points of `AB` and `BC`
`PQ` parallel to `AC` and `PQ=1/2AC` ..(ii) [Using the mid point theorem ]
(i)In `△ADC`,
`S` and `R` are the mid-point of `AD` and `DC`
`SR` parallel to `AC` and `SR=1/2AC` ...(ii) [ Using the mid point theorem ]
(iii)Using equation (i) and (ii), We get:
`PQ=SR` and `PQ` parallel to `SR`
(iv)`therefore PQRS` is a parallelogram.
Hence proved.