Let `square PQRS` be a quadrilateral. If M and N are the mid-points of the sides PQ and RS respectively, then PS+QR=

Options

A. `3overline(MN)`
B. `4overline(MN)`
C. `2overline(MN)`
D. `2overline(NM)`

Explanation

To solve the problem, we need to find the value of \( PS + QR \) given that \( M \) and \( N \) are the midpoints of sides \( PQ \) and \( RS \) respectively.

### Step-by-Step Solution:

1. **Define Position Vectors**:
Let the position vectors of points \( P, Q, R, S \) be represented as \( \vec{p}, \vec{q}, \vec{r}, \vec{s} \) respectively.

2. **Find Midpoints**:
Since \( M \) is the midpoint of \( PQ \), we can express its position vector as:
\[ \vec{m} = \frac{\vec{p} + \vec{q}}{2} \]
Similarly, since \( N \) is the midpoint of \( RS \), its position vector is:
\[ \vec{n} = \frac{\vec{r} + \vec{s}}{2} \]

3. **Express PS and QR**:
The vector \( PS \) can be expressed as:
\[ PS = \vec{s} - \vec{p} \]
The vector \( QR \) can be expressed as:
\[ QR = \vec{r} - \vec{q} \]

4. **Combine PS and QR**:
Now, we can add these two vectors:
\[ PS + QR = (\vec{s} - \vec{p}) + (\vec{r} - \vec{q}) = \vec{s} + \vec{r} - \vec{p} - \vec{q} \]

5. **Substitute Midpoint Expressions**:
From the midpoint definitions, we know:
\[ \vec{m} = \frac{\vec{p} + \vec{q}}{2} \quad \text{and} \quad \vec{n} = \frac{\vec{r} + \vec{s}}{2} \]
Rearranging gives:
\[ \vec{p} + \vec{q} = 2\vec{m} \quad \text{and} \quad \vec{r} + \vec{s} = 2\vec{n} \]

6. **Substituting into the Equation**:
Substitute these into the expression for \( PS + QR \):
\[ PS + QR = \vec{s} + \vec{r} - \vec{p} - \vec{q} = (2\vec{n}) - (2\vec{m}) = 2(\vec{n} - \vec{m}) \]

7. **Final Result**:
Thus, we have:
\[ PS + QR = 2(\vec{n} - \vec{m}) \]

### Conclusion:
The final answer is:
\[ PS + QR = 2(\vec{n} - \vec{m}) \]