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 Coordinates of Given Points:**
- \( A(-2, 3) \)
- \( B(6, 7) \)
- \( C(8, 3) \)

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

3. **Set Up the Equation for Midpoint \( E \) of Diagonal \( BD \):**
Since \( E \) is also the midpoint of diagonal \( BD \), we can express it as:
\[ E = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) \]
Substituting the coordinates of \( B(6, 7) \) and \( D(x, y) \):
\[ (3, 3) = \left( \frac{6 + x}{2}, \frac{7 + y}{2} \right) \]

4. **Create Two Equations from the Midpoint:**
From the x-coordinates:
\[ 3 = \frac{6 + x}{2} \]
Multiply both sides by 2:
\[ 6 = 6 + x \implies x = 0 \]

From the y-coordinates:
\[ 3 = \frac{7 + y}{2} \]
Multiply both sides by 2:
\[ 6 = 7 + y \implies y = 6 - 7 = -1 \]

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

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