Points A (1, 3) and C (5, 1) are opposite vertices of a rectangle ABCD. If the slope of BD is 2, then its equation is

Options

A. `2x-y=4`
B. `2x+y=4`
C. `2x+y-7=0`
D. `2x+y+7=0`

Answer

To find the equation of line BD given the points A(1, 3) and C(5, 1) as opposite vertices of rectangle ABCD and the slope of line BD as 2, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Midpoint of AC**:
The midpoint M of the line segment AC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
For points A(1, 3) and C(5, 1):
\[ M = \left( \frac{1 + 5}{2}, \frac{3 + 1}{2} \right) = \left( \frac{6}{2}, \frac{4}{2} \right) = (3, 2) \]

2. **Use the Slope to Write the Equation of Line BD**:
The slope \( m \) of line BD is given as 2. We can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Here, \( (x_1, y_1) \) is the midpoint M(3, 2) and \( m = 2 \):
\[ y - 2 = 2(x - 3) \]

3. **Simplify the Equation**:
Expanding the equation:
\[ y - 2 = 2x - 6 \]
Adding 2 to both sides:
\[ y = 2x - 4 \]

4. **Rearranging to Standard Form**:
To express the equation in standard form \( Ax + By + C = 0 \):
\[ 2x - y - 4 = 0 \]
This can also be rearranged to:
\[ 2x - y = 4 \]

### Final Answer
The equation of line BD is:
\[ 2x - y = 4 \]