1. **State the problem:** Find the equation of the perpendicular bisector of the line segment AB where A = (17,9) and B = (23,39). 2. **Formula and rules:** - The midpoint M of AB is given by $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$. - The slope of AB is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$. - The slope of the perpendicular bisector is the negative reciprocal of the slope of AB, i.e., $$m_{\perp} = -\frac{1}{m}$$. - The equation of a line with slope $m$ passing through point $(x_0,y_0)$ is $$y - y_0 = m(x - x_0)$$. 3. **Calculate midpoint M:** $$M = \left(\frac{17+23}{2}, \frac{9+39}{2}\right) = (20, 24)$$ 4. **Calculate slope of AB:** $$m = \frac{39 - 9}{23 - 17} = \frac{30}{6} = 5$$ 5. **Calculate slope of perpendicular bisector:** $$m_{\perp} = -\frac{1}{5}$$ 6. **Write equation of perpendicular bisector using point-slope form:** $$y - 24 = -\frac{1}{5}(x - 20)$$ 7. **Simplify to slope-intercept form $y = mx + c$:** $$y - 24 = -\frac{1}{5}x + \frac{20}{5}$$ $$y - 24 = -\frac{1}{5}x + 4$$ $$y = -\frac{1}{5}x + 4 + 24$$ $$y = -\frac{1}{5}x + 28$$ **Final answer:** $$y = -\frac{1}{5}x + 28$$