A(1, 3) and C(5, 1) are two opposite vertices of a rectangle ABCD. If the slope of BD is 2, then the coordinates of B can be :

Options

A. (4, 4)
B. (5, 4)
C. (2, 0)
D. (1, 0)

Answer

To find the coordinates of point B in rectangle ABCD with given vertices A(1, 3) and C(5, 1), and the slope of diagonal BD being 2, we can follow these steps:

### Step 1: Find the Midpoint of AC
The midpoint M of the diagonal 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):
\[ M = \left( \frac{1 + 5}{2}, \frac{3 + 1}{2} \right) = \left( \frac{6}{2}, \frac{4}{2} \right) = (3, 2) \]

### Step 2: Determine the Equation of Line BD
Since the slope of line BD is given as 2, we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using point M(3, 2) and slope m = 2:
\[ y - 2 = 2(x - 3) \]
Expanding this:
\[ y - 2 = 2x - 6 \implies y = 2x - 4 \]

### Step 3: Check Possible Coordinates for Point B
We need to find points B that satisfy the equation \(y = 2x - 4\). We can check several points:

1. **Point (4, 4)**:
\[ y = 2(4) - 4 = 8 - 4 = 4 \quad \text{(Valid)} \]

2. **Point (5, 4)**:
\[ y = 2(5) - 4 = 10 - 4 = 6 \quad \text{(Invalid)} \]

3. **Point (2, 0)**:
\[ y = 2(2) - 4 = 4 - 4 = 0 \quad \text{(Valid)} \]

4. **Point (1, 0)**:
\[ y = 2(1) - 4 = 2 - 4 = -2 \quad \text{(Invalid)} \]

### Conclusion
The possible coordinates for point B are:
- (4, 4)
- (2, 0)