Point A and B have co-ordinates (7, -3) and (1,9) respectively. Find :
the equation of perpendicular bisector of the line segment AB. 

Answer

To find the equation of the perpendicular bisector of the line segment AB, we will follow these steps:

### Step 1: Identify the coordinates of points A and B
- Point A has coordinates \( A(7, -3) \)
- Point B has coordinates \( B(1, 9) \)

### Step 2: Calculate the slope of line segment AB
The slope \( m_1 \) of line segment AB can be calculated using the formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m_1 = \frac{9 - (-3)}{1 - 7} = \frac{9 + 3}{1 - 7} = \frac{12}{-6} = -2 \]

### Step 3: Find the midpoint M of line segment AB
The midpoint \( M \) can be found 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:
\[ M = \left( \frac{7 + 1}{2}, \frac{-3 + 9}{2} \right) = \left( \frac{8}{2}, \frac{6}{2} \right) = (4, 3) \]

### Step 4: Determine the slope of the perpendicular bisector
The slope \( m_2 \) of the perpendicular bisector is the negative reciprocal of the slope of line segment AB:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2} \]

### Step 5: Use the point-slope form to find the equation of the perpendicular bisector
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting the midpoint \( M(4, 3) \) and the slope \( m_2 = \frac{1}{2} \):
\[ y - 3 = \frac{1}{2}(x - 4) \]

### Step 6: Simplify the equation
Distributing the slope:
\[ y - 3 = \frac{1}{2}x - 2 \]
Adding 3 to both sides:
\[ y = \frac{1}{2}x - 2 + 3 \]
\[ y = \frac{1}{2}x + 1 \]

### Final Equation
The equation of the perpendicular bisector of the line segment AB is:
\[ y = \frac{1}{2}x + 1 \]

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