If A(2 , 3) , B (1 , 4) , C(0 , -2) and D( x , y) are the vertices of a parallelogram , then what is the value of (x , y) ?

Options

A. (1 , -3)
B. (2 , 4)
C. (1 , 1)
D. (0 , 0)

Answer

To find the coordinates \( D(x, y) \) of the fourth vertex of the parallelogram formed by points \( A(2, 3) \), \( B(1, 4) \), and \( C(0, -2) \), we can use the property that the midpoints of the diagonals of a parallelogram are the same.

### Step-by-step Solution:

1. **Identify the midpoints of the diagonals**:
The diagonals of the parallelogram are \( AC \) and \( BD \). We need to find the midpoints of these diagonals.

2. **Calculate the midpoint of diagonal \( AC \)**:
The midpoint \( M_{AC} \) of points \( A(2, 3) \) and \( C(0, -2) \) can be calculated using the midpoint formula:
\[ M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 0}{2}, \frac{3 + (-2)}{2} \right) = \left( \frac{2}{2}, \frac{1}{2} \right) = (1, 0.5) \]

3. **Calculate the midpoint of diagonal \( BD \)**:
The midpoint \( M_{BD} \) of points \( B(1, 4) \) and \( D(x, y) \) is given by:
\[ M_{BD} = \left( \frac{1 + x}{2}, \frac{4 + y}{2} \right) \]

4. **Set the midpoints equal**:
Since the midpoints of the diagonals are equal, we can set the coordinates equal to each other:
\[ \frac{1 + x}{2} = 1 \quad \text{(1)} \]
\[ \frac{4 + y}{2} = 0.5 \quad \text{(2)} \]

5. **Solve for \( x \)**:
From equation (1):
\[ 1 + x = 2 \implies x = 2 - 1 = 1 \]

6. **Solve for \( y \)**:
From equation (2):
\[ 4 + y = 1 \implies y = 1 - 4 = -3 \]

7. **Conclusion**:
Thus, the coordinates of point \( D \) are \( D(1, -3) \).

### Final Answer:
The value of \( (x, y) \) is \( (1, -3) \).

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