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 three vertices, we can use the property that the diagonals of a parallelogram bisect each other.

Let's denote the points as follows:
- A = (-6, -4)
- B = (3, 5)
- C = (-2, 1)
- D = (x, y) (the fourth vertex we need to find)

### Step-by-Step Solution:

1. **Find the Midpoint of Diagonal AC:**
The midpoint \( M \) of diagonal AC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( A = (-6, -4) \) and \( C = (-2, 1) \).
\[ M_{AC} = \left( \frac{-6 + (-2)}{2}, \frac{-4 + 1}{2} \right) = \left( \frac{-8}{2}, \frac{-3}{2} \right) = (-4, -1.5) \]

2. **Find the Midpoint of Diagonal BD:**
The midpoint \( M \) of diagonal BD should be equal to the midpoint of diagonal AC. The coordinates of \( B \) are \( (3, 5) \) and \( D = (x, y) \).
\[ M_{BD} = \left( \frac{3 + x}{2}, \frac{5 + y}{2} \right) \]

3. **Set the Midpoints Equal:**
Since the midpoints must be equal, we set:
\[ \left( \frac{3 + x}{2}, \frac{5 + y}{2} \right) = (-4, -1.5) \]

4. **Create Two Equations:**
From the equality of the midpoints, we derive two equations:
\[ \frac{3 + x}{2} = -4 \quad \text{(1)} \]
\[ \frac{5 + y}{2} = -1.5 \quad \text{(2)} \]

5. **Solve Equation (1):**
Multiply both sides by 2:
\[ 3 + x = -8 \]
Subtract 3 from both sides:
\[ x = -8 - 3 = -11 \]

6. **Solve Equation (2):**
Multiply both sides by 2:
\[ 5 + y = -3 \]
Subtract 5 from both sides:
\[ y = -3 - 5 = -8 \]

7. **Conclusion:**
The coordinates of the fourth vertex \( D \) are:
\[ D = (x, y) = (-11, -8) \]

### Final Answer:
The fourth vertex of the parallelogram is \( (-11, -8) \).

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