Find the equation of prependicular bisector of the line segment joining the points `(4, -3) and (3, 1)`.

Answer

To find the equation of the perpendicular bisector of the line segment joining the points (4, -3) and (3, 1), we will follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( P \) of the line segment joining points \( A(4, -3) \) and \( B(3, 1) \) can be calculated using the midpoint formula:
\[ P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and B:
\[ P = \left( \frac{4 + 3}{2}, \frac{-3 + 1}{2} \right) = \left( \frac{7}{2}, \frac{-2}{2} \right) = \left( \frac{7}{2}, -1 \right) \]

### Step 2: Find the Slope of Line Segment AB
The slope \( m_1 \) of line segment \( AB \) can be calculated using the slope formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m_1 = \frac{1 - (-3)}{3 - 4} = \frac{1 + 3}{3 - 4} = \frac{4}{-1} = -4 \]

### Step 3: Find the Slope of the Perpendicular Bisector
Since the perpendicular bisector is perpendicular to line segment \( AB \), its slope \( m_2 \) can be found using the relationship:
\[ m_1 \cdot m_2 = -1 \]
Substituting \( m_1 = -4 \):
\[ -4 \cdot m_2 = -1 \implies m_2 = \frac{1}{4} \]

### Step 4: Use the Point-Slope Form to Find the Equation
Now we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using the midpoint \( P\left(\frac{7}{2}, -1\right) \) and slope \( m_2 = \frac{1}{4} \):
\[ y - (-1) = \frac{1}{4}\left(x - \frac{7}{2}\right) \]
This simplifies to:
\[ y + 1 = \frac{1}{4}x - \frac{7}{8} \]
Subtracting 1 from both sides:
\[ y = \frac{1}{4}x - \frac{7}{8} - 1 \]
Converting 1 to a fraction with a denominator of 8:
\[ y = \frac{1}{4}x - \frac{7}{8} - \frac{8}{8} = \frac{1}{4}x - \frac{15}{8} \]

### Step 5: Rearranging to Standard Form
To convert this to standard form \( Ax + By + C = 0 \):
\[ \frac{1}{4}x - y - \frac{15}{8} = 0 \]
Multiplying through by 8 to eliminate the fractions:
\[ 2x - 8y - 15 = 0 \]
Rearranging gives:
\[ 2x - 8y = 15 \]

### Final Answer
The equation of the perpendicular bisector of the line segment joining the points (4, -3) and (3, 1) is:
\[ 2x - 8y = 15 \]