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

Options

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

Answer

To find the coordinates of the fourth vertex of the parallelogram given the extremities of a diagonal and one vertex, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Let the points be:
- A (3, -4)
- B (-6, 5)
- C (-2, 1)
- D (unknown)

2. **Find the Midpoint of the Diagonal AB:**
- The midpoint \( O \) of the diagonal \( AB \) can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- Substituting the coordinates of points A and B:
\[ O = \left( \frac{3 + (-6)}{2}, \frac{-4 + 5}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) \]

3. **Set Up the Midpoint of the Other Diagonal (CD):**
- The midpoint of diagonal \( CD \) should also equal \( O \) since the diagonals of a parallelogram bisect each other.
- Let the coordinates of point D be \( (x, y) \). The midpoint of \( CD \) is given by:
\[ O = \left( \frac{-2 + x}{2}, \frac{1 + y}{2} \right) \]

4. **Equate the Midpoints:**
- Since both midpoints are equal, we can set up the following equations:
\[ \frac{-2 + x}{2} = \frac{-3}{2} \quad \text{(1)} \]
\[ \frac{1 + y}{2} = \frac{1}{2} \quad \text{(2)} \]

5. **Solve for x from Equation (1):**
- Multiply both sides by 2:
\[ -2 + x = -3 \]
- Rearranging gives:
\[ x = -3 + 2 = -1 \]

6. **Solve for y from Equation (2):**
- Multiply both sides by 2:
\[ 1 + y = 1 \]
- Rearranging gives:
\[ y = 1 - 1 = 0 \]

7. **Conclusion:**
- The coordinates of the fourth vertex \( D \) are \( (-1, 0) \).

### Final Answer:
The coordinates of the fourth vertex are \( D(-1, 0) \).