`A B C D` is a quadrilateral. `E` is the point of intersection of the line joining the midpoints of the opposite sides. If `O` is any point and ` vec O A+ vec O B+ vec O C+ vec O D=x vec O E ,t h e nx` is equal to a. `3` b. `9` c. `7` d. `4`

Explanation

We know that the figure formed by the lines joining the midpoints of the sides of a qudrilateral is a parallelogram. Hence, MPNQ is a parallelogramm, whose diagonals are MN and PQ intersecting at E, which is the midpoint of both MN and PQ. For any origin O, we have `vec(OA) + vec(OB) = 2(vec(OM))` (as M is the midpoint of AB)

`vec(OC) +vec(OB) = 2(vec(ON))` (as N is the midpoint of BC)
`rArr vec(OA) + vec(OB) + vec(OC) + vec(OD) = 2 (vec(OM) + vec(ON))`
`" " = 2(2 vec(OE)) = 4vec(OE)`
Where E is the midpoint of MN as it is the intersection of the diagonals of a parallelogram.