The three vertices of a parallelogram are A (3, - 4), B (-2, 1) and C (-6, 5). Which of the following cannot be the fourth one

Options

A. (-1, 0)
B. (7, -8)
C. (1, -5)
D. All of these

Answer

To determine which of the given points cannot be the fourth vertex of a parallelogram with the vertices A(3, -4), B(-2, 1), and C(-6, 5), we can use the property of the diagonals of a parallelogram. The diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Calculate the Midpoint of the Diagonal AC**:
The midpoint M of the diagonal AC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
For points A(3, -4) and C(-6, 5):
\[ M_{AC} = \left( \frac{3 + (-6)}{2}, \frac{-4 + 5}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) \]

2. **Calculate the Midpoint of the Diagonal BD**:
Let D be the fourth vertex of the parallelogram. The midpoint M of diagonal BD must also equal M_{AC}. Using the coordinates of B(-2, 1) and D(x, y):
\[ M_{BD} = \left( \frac{-2 + x}{2}, \frac{1 + y}{2} \right) \]
Setting this equal to M_{AC}:
\[ \left( \frac{-2 + x}{2}, \frac{1 + y}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) \]

3. **Set Up the Equations**:
From the equality of the midpoints, we can set up two equations:
- For the x-coordinates:
\[ \frac{-2 + x}{2} = \frac{-3}{2} \]
Multiplying both sides by 2:
\[ -2 + x = -3 \quad \Rightarrow \quad x = -1 \]
- For the y-coordinates:
\[ \frac{1 + y}{2} = \frac{1}{2} \]
Multiplying both sides by 2:
\[ 1 + y = 1 \quad \Rightarrow \quad y = 0 \]

4. **Identify the Fourth Vertex**:
Thus, the coordinates of the fourth vertex D are:
\[ D(-1, 0) \]

5. **Check the Given Options**:
Now we need to check which of the given options cannot be the fourth vertex. The options are:
- (-1, 0)
- (7, -8)
- (1, -5)
- All of these

Since we found that D(-1, 0) is indeed a valid vertex, we need to check the other options to see if they can form a parallelogram with the other three points.

6. **Check Other Points**:
- For (7, -8):
The midpoint with B(-2, 1) and (7, -8) does not equal the midpoint of AC.
- For (1, -5):
Similarly, the midpoint with B(-2, 1) and (1, -5) does not equal the midpoint of AC.

### Conclusion:
Therefore, the point that cannot be the fourth vertex of the parallelogram is **(7, -8)**.