The vertices of a parallelogram in order are A(1, 2), B(4, y), C(x, 6), D(3, 5), then (x, y) =

Options

A. (6, 3)
B. (3, 6)
C. (5, 6)
D. (1, 4)

Answer

To solve the problem of finding the coordinates (x, y) for the vertices of the parallelogram given the points A(1, 2), B(4, y), C(x, 6), and D(3, 5), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Midpoint of the Diagonals**:
The midpoints of the diagonals AC and BD must be equal.
- The midpoint of AC is given by:
\[ M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{1 + x}{2}, \frac{2 + 6}{2} \right) = \left( \frac{1 + x}{2}, 4 \right) \]
- The midpoint of BD is given by:
\[ M_{BD} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{4 + 3}{2}, \frac{y + 5}{2} \right) = \left( \frac{7}{2}, \frac{y + 5}{2} \right) \]

2. **Set the Midpoints Equal**:
Since the midpoints are equal, we can set the x-coordinates and y-coordinates equal to each other:
- For the x-coordinates:
\[ \frac{1 + x}{2} = \frac{7}{2} \]
- For the y-coordinates:
\[ 4 = \frac{y + 5}{2} \]

3. **Solve for x**:
From the equation \(\frac{1 + x}{2} = \frac{7}{2}\):
\[ 1 + x = 7 \implies x = 7 - 1 = 6 \]

4. **Solve for y**:
From the equation \(4 = \frac{y + 5}{2}\):
\[ 8 = y + 5 \implies y = 8 - 5 = 3 \]

5. **Final Coordinates**:
Thus, the coordinates (x, y) are:
\[ (x, y) = (6, 3) \]

### Conclusion:
The final answer is \((x, y) = (6, 3)\).