Find an equation for the perpendicular bisector of the line segment whose endpoints are \( (-2,2) \) and \( (-8,8) \). Answer Attempt 1 out of 2

To find the equation of the perpendicular bisector of the line segment with endpoints \( A(-2, 2) \) and \( B(-8, 8) \), we will follow these steps: 1. **Find the midpoint of the segment \( AB \)**. 2. **Calculate the slope of the line segment \( AB \)**. 3. **Determine the slope of the perpendicular bisector**. 4. **Use the point-slope form to write the equation of the perpendicular bisector**. ### Step 1: Find the Midpoint The midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] For our points \( A(-2, 2) \) and \( B(-8, 8) \): \[ M = \left( \frac{-2 + (-8)}{2}, \frac{2 + 8}{2} \right) = \left( \frac{-10}{2}, \frac{10}{2} \right) = (-5, 5) \] ### Step 2: Calculate the Slope of Line Segment \( AB \) The slope \( m \) of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For points \( A(-2, 2) \) and \( B(-8, 8) \): \[ m_{AB} = \frac{8 - 2}{-8 - (-2)} = \frac{6}{-6} = -1 \] ### Step 3: Determine the Slope of the Perpendicular Bisector The slope of the perpendicular bisector \( m_{\perp} \) is the negative reciprocal of the slope of \( AB \): \[ m_{\perp} = -\frac{1}{m_{AB}} = -\frac{1}{-1} = 1 \] ### Step 4: Use the Point-Slope Form The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Using the midpoint \( M(-5, 5) \) and the slope \( m_{\perp} = 1 \): \[ y - 5 = 1(x + 5) \] Simplifying this equation: \[ y - 5 = x + 5 \] \[ y = x + 10 \] Thus, the equation of the perpendicular bisector of the line segment whose endpoints are \( (-2, 2) \) and \( (-8, 8) \) is: \[ \boxed{y = x + 10} \]