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, given that A(1, 3) and C(5, 1) are opposite vertices and the slope of line BD is 2, we can follow these steps:

### Step 1: Find the midpoint O of diagonal AC
The midpoint O of the diagonal AC can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( A(1, 3) \) and \( C(5, 1) \).

Calculating:
\[ O = \left( \frac{1 + 5}{2}, \frac{3 + 1}{2} \right) = \left( \frac{6}{2}, \frac{4}{2} \right) = (3, 2) \]

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

### Step 3: Parameterize the line BD
Let \( B \) be represented as \( (t, 2t - 4) \), where \( t \) is a parameter. This gives us the coordinates of point B in terms of \( t \).

### Step 4: Determine the slope of AB
To ensure that AB and CD are perpendicular to BD, we need to calculate the slope of line AB:
\[ \text{slope of AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{(2t - 4) - 3}{t - 1} = \frac{2t - 7}{t - 1} \]

### Step 5: Determine the slope of CD
Let \( D \) be \( (x_D, y_D) \). The slope of line CD can be calculated similarly. Since D is opposite to B, we can express the coordinates of D in terms of the midpoint O:
\[ D = (6 - t, 4 - (2t - 4)) = (6 - t, 8 - 2t) \]
The slope of line CD is:
\[ \text{slope of CD} = \frac{(8 - 2t) - 1}{(6 - t) - 5} = \frac{7 - 2t}{1 - t} \]

### Step 6: Set the product of slopes to -1
For lines AB and CD to be perpendicular, the product of their slopes must equal -1:
\[ \left(\frac{2t - 7}{t - 1}\right) \cdot \left(\frac{7 - 2t}{1 - t}\right) = -1 \]
This simplifies to:
\[ (2t - 7)(7 - 2t) = -(t - 1)^2 \]

### Step 7: Solve the equation
Expanding both sides:
\[ 14t - 4t^2 - 49 + 14t = - (t^2 - 2t + 1) \]
Combining like terms:
\[ -4t^2 + 28t - 49 = -t^2 + 2t - 1 \]
Rearranging gives:
\[ -3t^2 + 26t - 48 = 0 \]

### Step 8: Factor the quadratic equation
Factoring:
\[ 3t^2 - 26t + 48 = 0 \]
Using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{26 \pm \sqrt{(-26)^2 - 4 \cdot 3 \cdot 48}}{2 \cdot 3} \]
Calculating the discriminant and solving gives us the possible values of \( t \).

### Step 9: Find coordinates of B
Substituting the values of \( t \) back into \( (t, 2t - 4) \) gives us the coordinates of B.

### Step 10: Verify the coordinates
Check if the calculated coordinates of B satisfy the conditions of forming a rectangle with vertices A, B, C, and D.