If `(-6,-4)` and ` (3,5)` are the extremities of the diagonal of a parallelogram and `(-2,1)` is its third vertex , then its fourth vertex is

Options

A. (-1,0)
B. (0,-1)
C. (-1,1)
D. none of these

Answer

To find the fourth vertex of the parallelogram given the coordinates of the other vertices, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Points:**
- The extremities of the diagonal are given as \( P(-6, -4) \) and \( R(3, 5) \).
- The third vertex \( Q \) is given as \( (-2, 1) \).
- We need to find the fourth vertex \( S(x, y) \).

2. **Use the Midpoint Formula:**
- The diagonals of a parallelogram bisect each other. Therefore, the midpoint of diagonal \( PR \) should be equal to the midpoint of diagonal \( QS \).
- The midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

3. **Calculate the Midpoint of \( PR \):**
- For points \( P(-6, -4) \) and \( R(3, 5) \):
\[ M_{PR} = \left( \frac{-6 + 3}{2}, \frac{-4 + 5}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) \]

4. **Set Up the Midpoint of \( QS \):**
- For points \( Q(-2, 1) \) and \( S(x, y) \):
\[ M_{QS} = \left( \frac{-2 + x}{2}, \frac{1 + y}{2} \right) \]

5. **Equate the Midpoints:**
- Since \( M_{PR} = M_{QS} \):
\[ \left( \frac{-3}{2}, \frac{1}{2} \right) = \left( \frac{-2 + x}{2}, \frac{1 + y}{2} \right) \]

6. **Set Up the Equations:**
- From the x-coordinates:
\[ \frac{-3}{2} = \frac{-2 + x}{2} \]
- From the y-coordinates:
\[ \frac{1}{2} = \frac{1 + y}{2} \]

7. **Solve for \( x \):**
- Multiply both sides of the x-coordinate equation by 2:
\[ -3 = -2 + x \]
- Rearranging gives:
\[ x = -3 + 2 = -1 \]

8. **Solve for \( y \):**
- Multiply both sides of the y-coordinate equation by 2:
\[ 1 = 1 + y \]
- Rearranging gives:
\[ y = 1 - 1 = 0 \]

9. **Conclusion:**
- The fourth vertex \( S \) is \( (-1, 0) \).

### Final Answer:
The fourth vertex of the parallelogram is \( S(-1, 0) \).