The coordinates of the points \( A \) and \( B \) are \( (-15,10) \) and \( (5,-6) \) respectively. Find the equation of the perpendicular bisector of \( A B \). Give your answer in the form \( a x+b y=c \) where \( a, b \) and \( c \) are integers in their lowest terms. .

To find the equation of the perpendicular bisector of the line segment joining points \( A(-15, 10) \) and \( B(5, -6) \), we follow these steps. 1. **Find the midpoint of \( AB \)**: \[ M = \left( \frac{-15 + 5}{2}, \frac{10 + (-6)}{2} \right) = \left( \frac{-10}{2}, \frac{4}{2} \right) = (-5, 2) \] 2. **Find the slope of line segment \( AB \)**: The slope \( m_{AB} \) is calculated as follows: \[ m_{AB} = \frac{-6 - 10}{5 - (-15)} = \frac{-16}{20} = -\frac{4}{5} \] 3. **Find the slope of the perpendicular bisector**: The slope of the perpendicular bisector \( m_{PB} \) is the negative reciprocal of \( m_{AB} \): \[ m_{PB} = \frac{5}{4} \] 4. **Use the point-slope form to find the equation**: Using the midpoint \( M(-5, 2) \) and the slope \( m_{PB} = \frac{5}{4} \): \[ y - 2 = \frac{5}{4}(x + 5) \] Rearranging gives: \[ y - 2 = \frac{5}{4}x + \frac{25}{4} \] \[ y = \frac{5}{4}x + \frac{25}{4} + 2 \] Converting \( 2 \) to a fraction gives: \[ y = \frac{5}{4}x + \frac{25}{4} + \frac{8}{4} = \frac{5}{4}x + \frac{33}{4} \] 5. **Convert to the form \( ax + by = c \)**: Multiply everything by \( 4 \) to eliminate the fractions: \[ 4y = 5x + 33 \implies -5x + 4y = 33 \] So, we can rearrange it as: \[ 5x - 4y = -33 \] The equation of the perpendicular bisector of \( AB \) in the form \( ax + by = c \) is: \[ 5x - 4y = -33 \]