The equation of the perpendicular bisector of the line segment joining A(2,-3) and B(-6,5) is

Options

A. x-y=-1
B. x-y=3
C. x+y=3
D. x+y=1

Explanation

To find the equation of the perpendicular bisector of the line segment joining points A(2, -3) and B(-6, 5), we will follow these steps:

### Step 1: Find the Midpoint of AB
The midpoint \( M \) of the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and B:
\[ M = \left( \frac{2 + (-6)}{2}, \frac{-3 + 5}{2} \right) = \left( \frac{-4}{2}, \frac{2}{2} \right) = (-2, 1) \]

### Step 2: Find the Slope of AB
The slope \( m_{AB} \) of the line segment AB is calculated using the formula:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m_{AB} = \frac{5 - (-3)}{-6 - 2} = \frac{5 + 3}{-8} = \frac{8}{-8} = -1 \]

### Step 3: Find the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector \( m_{CD} \) is the negative reciprocal of the slope of AB:
\[ m_{CD} = -\frac{1}{m_{AB}} = -\frac{1}{-1} = 1 \]

### Step 4: Use the Point-Slope Form to Find the Equation
We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using the midpoint \( M(-2, 1) \) and the slope \( m_{CD} = 1 \):
\[ y - 1 = 1(x + 2) \]
Simplifying this:
\[ y - 1 = x + 2 \]
\[ y = x + 3 \]

### Final Equation
The equation of the perpendicular bisector of the line segment joining A(2, -3) and B(-6, 5) is:
\[ y = x + 3 \]

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