The fourth vertex D of a parallelogram ABCD whose three vertices are A (-2,3), B (6,7) and C (8,3) is

Options

A. (0,1)
B. (0,-1)
C. (-1,0)
D. (1,0)

Explanation

To find the fourth vertex \( D \) of the parallelogram \( ABCD \) given the vertices \( A(-2, 3) \), \( B(6, 7) \), and \( C(8, 3) \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Midpoint of Diagonal AC:**
We first find the midpoint \( O \) of diagonal \( AC \). The coordinates of \( A \) and \( C \) are:
- \( A(-2, 3) \)
- \( C(8, 3) \)

Using the midpoint formula:
\[ O_x = \frac{x_1 + x_2}{2} = \frac{-2 + 8}{2} = \frac{6}{2} = 3 \]
\[ O_y = \frac{y_1 + y_2}{2} = \frac{3 + 3}{2} = \frac{6}{2} = 3 \]
Thus, the coordinates of midpoint \( O \) are \( (3, 3) \).

2. **Set Up the Midpoint of Diagonal BD:**
Next, we express the coordinates of midpoint \( O \) in terms of the unknown vertex \( D(x_4, y_4) \) and vertex \( B(6, 7) \):
\[ O_x = \frac{x_4 + 6}{2} \]
\[ O_y = \frac{y_4 + 7}{2} \]

3. **Equate the Midpoints:**
Since both expressions represent the same point \( O(3, 3) \), we can set up the following equations:
\[ \frac{x_4 + 6}{2} = 3 \]
\[ \frac{y_4 + 7}{2} = 3 \]

4. **Solve for \( x_4 \):**
From the first equation:
\[ x_4 + 6 = 6 \quad \text{(Multiplying both sides by 2)} \]
\[ x_4 = 6 - 6 = 0 \]

5. **Solve for \( y_4 \):**
From the second equation:
\[ y_4 + 7 = 6 \quad \text{(Multiplying both sides by 2)} \]
\[ y_4 = 6 - 7 = -1 \]

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

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