Find the equation of perpendicular bisector of line segment joining the points `(1,5)` and `(3,-1)`

Answer

To find the equation of the perpendicular bisector of the line segment joining the points \( A(1, 5) \) and \( B(3, -1) \), we will follow these steps:

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

Calculating the midpoint:
\[ M = \left( \frac{1 + 3}{2}, \frac{5 + (-1)}{2} \right) = \left( \frac{4}{2}, \frac{4}{2} \right) = (2, 2) \]

### Step 2: Find the Slope of Line Segment AB
The slope \( m \) of the line segment joining two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Calculating the slope for points \( A(1, 5) \) and \( B(3, -1) \):
\[ m_{AB} = \frac{-1 - 5}{3 - 1} = \frac{-6}{2} = -3 \]

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

### 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 \( M(2, 2) \) and the slope \( m_{PB} = \frac{1}{3} \):
\[ y - 2 = \frac{1}{3}(x - 2) \]

### Step 5: Simplify the Equation
Expanding the equation:
\[ y - 2 = \frac{1}{3}x - \frac{2}{3} \]
Adding \( 2 \) to both sides:
\[ y = \frac{1}{3}x - \frac{2}{3} + 2 \]
Converting \( 2 \) to a fraction with a denominator of 3:
\[ y = \frac{1}{3}x - \frac{2}{3} + \frac{6}{3} = \frac{1}{3}x + \frac{4}{3} \]

### Step 6: Convert to Standard Form
To convert to standard form \( Ax + By + C = 0 \):
\[ 3y = x + 4 \quad \Rightarrow \quad x - 3y + 4 = 0 \]

### Final Answer
The equation of the perpendicular bisector is:
\[ \boxed{x - 3y + 4 = 0} \]