The vertices of a parallelogram in order are `A(1,2), B (4,y), C (x,6)` and `D(3,5)` . Then `(x,y)` is :

Options

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

Answer

To find the values of \(x\) and \(y\) for the vertices of the parallelogram \(A(1,2)\), \(B(4,y)\), \(C(x,6)\), 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 Midpoints of the Diagonals:**
The midpoints of the diagonals \(AC\) and \(BD\) must be equal.
- The midpoint \(O\) of diagonal \(AC\) can be calculated as:
\[ O_{AC} = \left( \frac{1 + x}{2}, \frac{2 + 6}{2} \right) = \left( \frac{1 + x}{2}, 4 \right) \]
- The midpoint \(O\) of diagonal \(BD\) can be calculated as:
\[ O_{BD} = \left( \frac{4 + 3}{2}, \frac{y + 5}{2} \right) = \left( \frac{7}{2}, \frac{y + 5}{2} \right) \]

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 \( \frac{1 + x}{2} = \frac{7}{2} \):
\[ 1 + x = 7 \]
\[ x = 7 - 1 = 6 \]

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

5. **Final Result:**
The values of \(x\) and \(y\) are:
\[ (x, y) = (6, 3) \]

### Conclusion:
Thus, the coordinates \((x, y)\) are \((6, 3)\).