Given two points A (-5, 2) and B (1, -4), find :
 equation of the perpendicular bisector of AB.

Answer

To find the equation of the perpendicular bisector of the line segment joining points A (-5, 2) and B (1, -4), we will follow these steps:

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

### Step 2: Find the Slope of AB
The slope \( m_1 \) of the line segment AB is calculated using the formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m_1 = \frac{-4 - 2}{1 - (-5)} = \frac{-6}{6} = -1 \]

### Step 3: Find the Slope of the Perpendicular Bisector
The slope \( m_2 \) of the perpendicular bisector is the negative reciprocal of \( m_1 \):
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-1} = 1 \]

### Step 4: Use the Point-Slope Form to Find the Equation
The equation of a line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Using the midpoint \( M(-2, -1) \) and the slope \( m_2 = 1 \):
\[ y - (-1) = 1(x - (-2)) \]
This simplifies to:
\[ y + 1 = x + 2 \]

### Step 5: Rearranging to Standard Form
Rearranging the equation:
\[ y + 1 - x - 2 = 0 \]
This simplifies to:
\[ -x + y - 1 = 0 \quad \text{or} \quad x - y + 1 = 0 \]

### Final Answer
The equation of the perpendicular bisector of line segment AB is:
\[ x - y + 1 = 0 \]

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