A(-1,1),B(5,3) are opposite vertices of a square in xy-plane.
The equation of the other diagonal (not passing throug A,B) of the square is given by

Options

A. `x-3y+4=0`
B. `2x-y+3=0`
C. `y+3x-8=0`
D. `x+2y-1=0`

Answer

To find the equation of the other diagonal (not passing through points A and B) of the square with vertices A(-1, 1) and B(5, 3), we can follow these steps:

### Step 1: Find the Midpoint of AB
The midpoint \( M \) of the line segment connecting points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and B:
\[ M = \left( \frac{-1 + 5}{2}, \frac{1 + 3}{2} \right) = \left( \frac{4}{2}, \frac{4}{2} \right) = (2, 2) \]

### Step 2: Calculate the Slope of AB
The slope \( m_{AB} \) of the line segment AB can be calculated using the formula:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A and B:
\[ m_{AB} = \frac{3 - 1}{5 - (-1)} = \frac{2}{6} = \frac{1}{3} \]

### Step 3: Determine the Slope of the Other Diagonal (CD)
Since the diagonals of a square are perpendicular, the slope \( m_{CD} \) of the diagonal CD will be the negative reciprocal of \( m_{AB} \):
\[ m_{CD} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{1}{3}} = -3 \]

### Step 4: Use the Point-Slope Form to Find the Equation of CD
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 \( M(2, 2) \) and \( m = -3 \):
\[ y - 2 = -3(x - 2) \]
Expanding this:
\[ y - 2 = -3x + 6 \]
\[ y = -3x + 8 \]

### Step 5: Rearranging to Standard Form
Rearranging the equation to standard form:
\[ 3x + y - 8 = 0 \]

Thus, the equation of the other diagonal (not passing through A and B) of the square is:
\[ 3x + y - 8 = 0 \]