1. **State the problem:** Find the equation of the perpendicular bisector of the line segment joining points $A(0,6)$ and $B(2,-2)$.\n\n2. **Find the midpoint of segment AB:** The midpoint $M$ is given by $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) = \left(\frac{0+2}{2}, \frac{6+(-2)}{2}\right) = (1, 2).$$\n\n3. **Calculate the slope of segment AB:** The slope $m_{AB}$ is $$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 6}{2 - 0} = \frac{-8}{2} = -4.$$\n\n4. **Find the slope of the perpendicular bisector:** The perpendicular bisector is perpendicular to AB, so its slope $m_{\perp}$ is the negative reciprocal of $m_{AB}$: $$m_{\perp} = -\frac{1}{m_{AB}} = -\frac{1}{-4} = \frac{1}{4}.$$\n\n5. **Write the equation of the perpendicular bisector:** Using point-slope form with point $M(1,2)$ and slope $\frac{1}{4}$: $$y - 2 = \frac{1}{4}(x - 1).$$\n\n6. **Simplify the equation:**\n$$y - 2 = \frac{1}{4}x - \frac{1}{4}$$\n$$y = \frac{1}{4}x - \frac{1}{4} + 2 = \frac{1}{4}x + \frac{7}{4}.$$\n\n**Final answer:** The equation of the perpendicular bisector is $$y = \frac{1}{4}x + \frac{7}{4}.$$