If P and Q are the middle points of the sides BC and CD of the parallelogram ABCD, then AP+AQ is equal to

Options

A. AC
B. `(1)/(2)AC`
C. `(2)/(3)AC`
D. `(3)/(2)AC`

Explanation

To solve the problem, we need to find the sum of the vectors \( \vec{AP} + \vec{AQ} \) where \( P \) and \( Q \) are the midpoints of sides \( BC \) and \( CD \) of the parallelogram \( ABCD \).

### Step-by-Step Solution:

1. **Identify Points and Vectors:**
Let the position vectors of points \( A, B, C, D \) be represented as \( \vec{A}, \vec{B}, \vec{C}, \vec{D} \) respectively. Since \( ABCD \) is a parallelogram, we have:
\[ \vec{C} = \vec{A} + \vec{B} - \vec{D} \]

2. **Find Midpoints \( P \) and \( Q \):**
The midpoint \( P \) of side \( BC \) can be expressed as:
\[ \vec{P} = \frac{\vec{B} + \vec{C}}{2} \]
The midpoint \( Q \) of side \( CD \) can be expressed as:
\[ \vec{Q} = \frac{\vec{C} + \vec{D}}{2} \]

3. **Express \( \vec{AP} \) and \( \vec{AQ} \):**
The vector \( \vec{AP} \) is given by:
\[ \vec{AP} = \vec{P} - \vec{A} = \left( \frac{\vec{B} + \vec{C}}{2} \right) - \vec{A} \]
The vector \( \vec{AQ} \) is given by:
\[ \vec{AQ} = \vec{Q} - \vec{A} = \left( \frac{\vec{C} + \vec{D}}{2} \right) - \vec{A} \]

4. **Substituting \( \vec{C} \):**
Substitute \( \vec{C} = \vec{A} + \vec{B} - \vec{D} \) into the equations for \( \vec{AP} \) and \( \vec{AQ} \):
\[ \vec{AP} = \frac{\vec{B} + (\vec{A} + \vec{B} - \vec{D})}{2} - \vec{A} = \frac{2\vec{B} + \vec{A} - \vec{D}}{2} - \vec{A} = \frac{2\vec{B} - \vec{A} - \vec{D}}{2} \]
\[ \vec{AQ} = \frac{(\vec{A} + \vec{B} - \vec{D}) + \vec{D}}{2} - \vec{A} = \frac{\vec{A} + \vec{B}}{2} - \vec{A} = \frac{\vec{B} - \vec{A}}{2} \]

5. **Adding \( \vec{AP} \) and \( \vec{AQ} \):**
Now, we can add \( \vec{AP} \) and \( \vec{AQ} \):
\[ \vec{AP} + \vec{AQ} = \left( \frac{2\vec{B} - \vec{A} - \vec{D}}{2} \right) + \left( \frac{\vec{B} - \vec{A}}{2} \right) \]
Combine the terms:
\[ = \frac{(2\vec{B} - \vec{A} - \vec{D}) + (\vec{B} - \vec{A})}{2} = \frac{3\vec{B} - 2\vec{A} - \vec{D}}{2} \]

6. **Expressing in terms of \( \vec{AC} \):**
Since \( \vec{AC} = \vec{C} - \vec{A} \):
\[ \vec{AC} = \vec{A} + \vec{B} - \vec{D} - \vec{A} = \vec{B} - \vec{D} \]
Therefore, we can express \( \vec{AP} + \vec{AQ} \) in terms of \( \vec{AC} \):
\[ \vec{AP} + \vec{AQ} = \frac{3}{2} \vec{AC} \]

### Conclusion:
Thus, we find that:
\[ \vec{AP} + \vec{AQ} = \frac{3}{2} \vec{AC} \]