`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 `OA=a,OB=b,OC=c`
and OD=d
Therefore, `OA+OB+OC+OD`
`=a+b+c+d`
P, the mid-point of `AB`, is `(a+b)/(2)`
Q, the mid-point of Cd, is `(c+d)/(2)`.
therefore, the mid-point of PQ is `(a+b+c+d)/(4)`
Similarly, the mid point of RS is `(a+b+c+d)/(4)`
i.e., `OE=(a+b+c+d)/4`.
`implies x=4`.