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The problem is asking for the equation of the perpendicular bisector of line segment \(AB\), given points \(A(-2, 7)\) and \(B(-6, -1)\).

### Solution Steps

1. **Find the Midpoint of \(AB\):**

   The midpoint \(M\) of a line segment with endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:
   \[
   M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   For points \(A(-2, 7)\) and \(B(-6, -1)\):
   \[
   M = \left( \frac{-2 + (-6)}{2}, \frac{7 + (-1)}{2} \right) = \left( \frac{-8}{2}, \frac{6}{2} \right) = (-4, 3)
   \]

2. **Calculate the Slope of \(AB\):**

   The slope \(m_{AB}\) of a line passing through points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is:
   \[
   m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   For points \(A(-2, 7)\) and \(B(-6, -1)\):
   \[
   m_{AB} = \frac{-1 - 7}{-6 - (-2)} = \frac{-8}{-4} = 2
   \]

3. **Determine the Slope of the Perpendicular Bisector:**

   The perpendicular bisector of \(AB\) will have a slope that is the negative reciprocal of \(m_{AB}\). Thus, if \(m_{AB} = 2\), the perpendicular slope \(m_{\perp}\) is:
   \[
   m_{\perp} = -\frac{1}{2}
   \]

4. **Write the Equation of the Perpendicular Bisector:**

   The equation of a line with slope \(m\) passing through a point \((x_1, y_1)\) is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \(m = -\frac{1}{2}\) and the midpoint \((-4, 3)\):
   \[
   y - 3 = -\frac{1}{2}(x + 4)
   \]

5. **Simplify the Equation:**

   Distribute and simplify:
   \[
   y - 3 = -\frac{1}{2}x - 2
   \]
   \[
   y = -\frac{1}{2}x + 1
   \]

### Final Answer
The equation of the perpendicular bisector of \(AB\) is:
\[
y = -\frac{1}{2}x + 1
\]

Would you like further clarification on any step?

### Related Questions
1. How do you find the midpoint of a line segment in 3D space?
2. How can you derive the formula for the perpendicular slope?
3. What if the line segment is vertical? How do you handle the perpendicular bisector?
4. How can this method be extended to find the perpendicular bisector in 3D?
5. How do you find the equation of the line if the slope is undefined?

### Tip
Always double-check your slopes and midpoint calculations as they are critical to accurately finding the perpendicular bisector.

已知两点 A(-2, 7) 及 B(-6, -1)，求 AB 的垂直平分线的方程式。

Discover the solution to finding the perpendicular bisector of line segment AB given points A(-2,7) and B(-6,-1). This problem involves calculating the midpoint and slopes to derive the equation of the bisector, ideal for students in Grades 10-11.

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