`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`

Options

A. 3
B. 9
C. 7
D. 4

Answer

D

Explanation

Let `vec(OA) = veca, vec(OB) = vecb, vec(OC)= vecc and vec(OD) = vecd`.
Therefore,
`" "vec(OA) + vec(OB) + vec(OC) + vec(OD) = veca + vecb +vecc + vecd`
`P`, the midpoint of AB, is `(veca+vecb)/(2)`
`Q`, the midpoint of `vec(CD)`, is `(vecc + vecd)/(2)`
Therfore, the midpoint of `vec(PQ)` is `(veca + vecb+vecc +vecd)/(4)`
Similarly, the midpoint of RS is `(veca+vecb+vecc+vecd)/(4) rArr x = 4`.