If the points A(7,3) and C(0,-4) are two opposite vertices of a rhombus ABCD, then the equation of the diagonal BD is

Options

A. `x-y-3=0`
B. `x+y+3=0`
C. `x-y+3=0`
D. `x+y-4=0`

Answer

To find the equation of the diagonal BD of the rhombus ABCD, where the opposite vertices A(7, 3) and C(0, -4) are given, we can follow these steps:

### Step 1: Find the midpoint E of diagonal AC
The midpoint E of the line segment AC can be calculated using the midpoint formula:
\[ E = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( A(7, 3) \) and \( C(0, -4) \).

Calculating the coordinates of E:
\[ E_x = \frac{7 + 0}{2} = \frac{7}{2} \]
\[ E_y = \frac{3 + (-4)}{2} = \frac{3 - 4}{2} = \frac{-1}{2} \]
Thus, the coordinates of E are \( E\left(\frac{7}{2}, -\frac{1}{2}\right) \).

### Step 2: Find the slope of diagonal AC
Next, we need to find the slope of line AC. The slope \( m \) of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points A(7, 3) and C(0, -4):
\[ m_{AC} = \frac{-4 - 3}{0 - 7} = \frac{-7}{-7} = 1 \]

### Step 3: Find the slope of diagonal BD
Since the diagonals of a rhombus bisect each other at right angles, the slope of diagonal BD will be the negative reciprocal of the slope of AC:
\[ m_{BD} = -\frac{1}{m_{AC}} = -\frac{1}{1} = -1 \]

### Step 4: Use point-slope form to find the equation of line BD
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the midpoint E \( \left(\frac{7}{2}, -\frac{1}{2}\right) \) and \( m = -1 \):
\[ y - \left(-\frac{1}{2}\right) = -1\left(x - \frac{7}{2}\right) \]
Simplifying this:
\[ y + \frac{1}{2} = -x + \frac{7}{2} \]
\[ y = -x + \frac{7}{2} - \frac{1}{2} \]
\[ y = -x + 3 \]

### Step 5: Rearranging to standard form
Rearranging the equation to standard form:
\[ x + y - 3 = 0 \]
or equivalently:
\[ x + y - 3 = 0 \]

### Final Answer
Thus, the equation of the diagonal BD is:
\[ x + y - 3 = 0 \]