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

Answer

To find the equation of the diagonal \( AD \) in parallelogram \( ABDC \) given the coordinates of points \( A \), \( B \), and \( C \), we can follow these steps:

### Step 1: Identify the coordinates
The coordinates of the points are:
- \( A(1, 2) \)
- \( B(3, 4) \)
- \( C(2, 5) \)

### Step 2: Find the midpoint of diagonal \( BC \)
The midpoint \( O \) of diagonal \( BC \) can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of \( B \) and \( C \):
\[ O = \left( \frac{3 + 2}{2}, \frac{4 + 5}{2} \right) = \left( \frac{5}{2}, \frac{9}{2} \right) \]

### Step 3: Use the midpoint to find coordinates of point \( D \)
Since \( O \) is also the midpoint of diagonal \( AD \), we can set up the following equations:
\[ \frac{x_A + x_D}{2} = \frac{5}{2} \quad \text{and} \quad \frac{y_A + y_D}{2} = \frac{9}{2} \]
Substituting the coordinates of \( A(1, 2) \):
1. For \( x \):
\[ \frac{1 + x_D}{2} = \frac{5}{2} \]
Multiplying both sides by 2:
\[ 1 + x_D = 5 \implies x_D = 4 \]

2. For \( y \):
\[ \frac{2 + y_D}{2} = \frac{9}{2} \]
Multiplying both sides by 2:
\[ 2 + y_D = 9 \implies y_D = 7 \]

Thus, the coordinates of point \( D \) are \( D(4, 7) \).

### Step 4: Find the equation of line \( AD \)
We can use the two-point form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope of the line. The slope \( m \) between points \( A(1, 2) \) and \( D(4, 7) \) is calculated as:
\[ m = \frac{y_D - y_A}{x_D - x_A} = \frac{7 - 2}{4 - 1} = \frac{5}{3} \]

Now, substituting \( A(1, 2) \) into the equation:
\[ y - 2 = \frac{5}{3}(x - 1) \]

### Step 5: Rearranging to standard form
Multiplying through by 3 to eliminate the fraction:
\[ 3(y - 2) = 5(x - 1) \]
Expanding both sides:
\[ 3y - 6 = 5x - 5 \]
Rearranging gives:
\[ 5x - 3y + 1 = 0 \]

Thus, the equation of diagonal \( AD \) is:
\[ \boxed{5x - 3y + 1 = 0} \]