If in parallelogram `ABDC`, the coordinate of `A`, `B` and `C` are respectively `(1,2)`, `(3,4)` and `(2,5)`, then the equation of the diagonal `AD` is

Options

A. `3x+5y-13=0`
B. `3x-5y+7=0`
C. `5x-3y+1=0`
D. `5x+3y-11=0`

Explanation

To find the equation of diagonal \( AD \) in the parallelogram \( ABCD \) with given coordinates for points \( A(1, 2) \), \( B(3, 4) \), and \( C(2, 5) \), we can follow these steps:

### Step 1: Find the Coordinates of Point D
In a parallelogram, the diagonals bisect each other. Therefore, the midpoint \( P \) of diagonal \( AC \) will also be the midpoint of diagonal \( BD \).

1. **Calculate the midpoint \( P \) of \( AC \)**:
\[ P = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{1 + 2}{2}, \frac{2 + 5}{2} \right) = \left( \frac{3}{2}, \frac{7}{2} \right) \]

2. **Let the coordinates of point \( D \) be \( (x_D, y_D) \)**. Since \( P \) is also the midpoint of \( BD \):
\[ P = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) \]

Setting the coordinates equal gives us two equations:
\[ \frac{3 + x_D}{2} = \frac{3}{2} \quad \text{and} \quad \frac{4 + y_D}{2} = \frac{7}{2} \]

3. **Solve for \( x_D \) and \( y_D \)**:
- From the first equation:
\[ 3 + x_D = 3 \implies x_D = 0 \]
- From the second equation:
\[ 4 + y_D = 7 \implies y_D = 3 \]

Thus, the coordinates of point \( D \) are \( (0, 3) \).

### Step 2: Find the Equation of Line \( AD \)
Now that we have the coordinates of points \( A(1, 2) \) and \( D(0, 3) \), we can find the equation of line \( AD \).

1. **Calculate the slope \( m \) of line \( AD \)**:
\[ m = \frac{y_D - y_A}{x_D - x_A} = \frac{3 - 2}{0 - 1} = \frac{1}{-1} = -1 \]

2. **Use point-slope form to write the equation of line \( AD \)**:
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using point \( A(1, 2) \):
\[ y - 2 = -1(x - 1) \]

3. **Simplify the equation**:
\[ y - 2 = -x + 1 \implies y = -x + 3 \]

4. **Rearranging to standard form**:
\[ x + y - 3 = 0 \]

### Final Equation of Diagonal \( AD \)
The equation of diagonal \( AD \) is:
\[ x + y - 3 = 0 \]