`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

Explanation

To solve the problem, we need to find the value of \( x \) in the equation:

\[ \vec{O A} + \vec{O B} + \vec{O C} + \vec{O D} = x \vec{O E} \]

where \( E \) is the point of intersection of the lines joining the midpoints of the opposite sides of quadrilateral \( ABCD \).

### Step 1: Define the midpoints
Let \( P, Q, R, S \) be the midpoints of sides \( AB, BC, CD, \) and \( DA \) respectively. The coordinates of these midpoints can be expressed as follows:

- \( \vec{P} = \frac{\vec{A} + \vec{B}}{2} \)
- \( \vec{Q} = \frac{\vec{B} + \vec{C}}{2} \)
- \( \vec{R} = \frac{\vec{C} + \vec{D}}{2} \)
- \( \vec{S} = \frac{\vec{D} + \vec{A}}{2} \)

### Step 2: Identify point \( E \)
Point \( E \) is the intersection of lines \( PR \) and \( QS \). We can express \( \vec{E} \) in terms of the midpoints.

Using the section formula, if \( E \) divides \( PR \) in the ratio \( 1:1 \) (since \( E \) is the midpoint of \( PR \)), we can write:

\[ \vec{E} = \frac{\vec{P} + \vec{R}}{2} = \frac{\frac{\vec{A} + \vec{B}}{2} + \frac{\vec{C} + \vec{D}}{2}}{2} \]

Simplifying this gives:

\[ \vec{E} = \frac{\vec{A} + \vec{B} + \vec{C} + \vec{D}}{4} \]

### Step 3: Substitute \( \vec{E} \) into the equation
Now we substitute \( \vec{E} \) back into the original equation:

\[ \vec{O A} + \vec{O B} + \vec{O C} + \vec{O D} = x \cdot \frac{\vec{A} + \vec{B} + \vec{C} + \vec{D}}{4} \]

### Step 4: Express the left-hand side
The left-hand side can be expressed as:

\[ \vec{O A} + \vec{O B} + \vec{O C} + \vec{O D} = \vec{A} + \vec{B} + \vec{C} + \vec{D} \]

### Step 5: Set the equations equal
Now we equate both sides:

\[ \vec{A} + \vec{B} + \vec{C} + \vec{D} = x \cdot \frac{\vec{A} + \vec{B} + \vec{C} + \vec{D}}{4} \]

### Step 6: Solve for \( x \)
Assuming \( \vec{A} + \vec{B} + \vec{C} + \vec{D} \neq 0 \), we can divide both sides by \( \vec{A} + \vec{B} + \vec{C} + \vec{D} \):

\[ 1 = \frac{x}{4} \]

Multiplying both sides by 4 gives:

\[ x = 4 \]

### Conclusion
Thus, the value of \( x \) is:

\[ \boxed{4} \]