Which of the following statements are correct :- If M is the mid point of AB and O is any point, then

Options

A. `OM=OA+MA`
B. `OM=OA-MA`
C. `OM=1/2(OA-OB)`
D. `OM=1/2(OB+OA)`

Answer

To solve the problem, we need to analyze the situation where M is the midpoint of segment AB, and O is any point in space. We will derive the relationship between the vectors involved.

### Step-by-Step Solution:

1. **Understanding the Midpoint**:
Since M is the midpoint of segment AB, we can express the relationship between the vectors as:
\[ \vec{AM} = \vec{MB} \]
This means that the vector from A to M is equal to the vector from M to B.

2. **Using Vector Addition**:
By the triangle law of vector addition, we can say:
\[ \vec{OA} + \vec{AB} = \vec{OB} \]
Rearranging this gives us:
\[ \vec{AB} = \vec{OB} - \vec{OA} \]

3. **Expressing Vector AM**:
Since M is the midpoint, we can express vector AM in terms of OA and OB:
\[ \vec{AM} = \frac{1}{2} \vec{AB} \]
Substituting the expression for AB:
\[ \vec{AM} = \frac{1}{2} (\vec{OB} - \vec{OA}) \]

4. **Expressing Vector OM**:
Now, we can express vector OM in terms of OA and OB. We know:
\[ \vec{OM} = \vec{OA} + \vec{AM} \]
Substituting AM:
\[ \vec{OM} = \vec{OA} + \frac{1}{2} (\vec{OB} - \vec{OA}) \]

5. **Simplifying the Expression**:
Now we simplify the equation:
\[ \vec{OM} = \vec{OA} + \frac{1}{2} \vec{OB} - \frac{1}{2} \vec{OA} \]
\[ \vec{OM} = \frac{1}{2} \vec{OA} + \frac{1}{2} \vec{OB} \]

6. **Final Result**:
Thus, we can conclude:
\[ \vec{OM} = \frac{1}{2} (\vec{OA} + \vec{OB}) \]

### Conclusion:
The correct statement from the options provided is:
\[ \vec{OM} = \frac{1}{2} (\vec{OA} + \vec{OB}) \]
This corresponds to option D.