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We need to find the equation for the perpendicular bisector of line segment \(AC\) in the triangle \( \triangle ABC \), where \( A(-1,1) \) and \( C(1,6) \). Let's break this into steps:

### Step 1: Find the Midpoint of \(AC\)
The midpoint \(M\) of a line segment with endpoints \(A(x_1, y_1)\) and \(C(x_2, y_2)\) can be calculated using the midpoint formula:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

For \(A(-1, 1)\) and \(C(1, 6)\), we calculate:

\[
M = \left( \frac{-1 + 1}{2}, \frac{1 + 6}{2} \right) = (0, 3.5)
\]

Thus, the midpoint \(M\) is \( (0, 3.5) \).

### Step 2: Find the Slope of Line \(AC\)
The slope \(m\) of a line passing through two 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 \(A(-1, 1)\) and \(C(1, 6)\), we calculate:

\[
m_{AC} = \frac{6 - 1}{1 - (-1)} = \frac{5}{2}
\]

Thus, the slope of line \(AC\) is \( \frac{5}{2} \).

### Step 3: Find the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of line \(AC\). If the slope of \(AC\) is \( \frac{5}{2} \), the slope of the perpendicular bisector will be:

\[
m_{\text{perpendicular}} = -\frac{2}{5}
\]

### Step 4: Write the Equation of the Perpendicular Bisector
We now use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

where \(m\) is the slope and \((x_1, y_1)\) is the midpoint \(M(0, 3.5)\). Substituting the values:

\[
y - 3.5 = -\frac{2}{5}(x - 0)
\]

Simplifying:

\[
y - 3.5 = -\frac{2}{5}x
\]

\[
y = -\frac{2}{5}x + 3.5
\]

Thus, the equation of the perpendicular bisector of line segment \(AC\) is:

\[
y = -\frac{2}{5}x + 3.5
\]

Would you like to explore more details or clarify anything?

### Related Questions:
1. How do you find the equation of a line using two points?
2. What is the significance of the perpendicular bisector in a triangle?
3. Can you find the equation of the line parallel to \(AC\) passing through a specific point?
4. How does changing the coordinates of points \(A\) or \(C\) affect the slope of \(AC\)?
5. How do you calculate the length of line segment \(AC\)?

### Tip:
For perpendicular bisectors, always remember to calculate the slope of the original line and then use its negative reciprocal to find the perpendicular slope.

Find the equation for the perpendicular bisector of AC in triangle ABC with points A(-1,1), B(-4,5), and C(1,6).

Explore the step-by-step process of finding the equation of the perpendicular bisector of line segment AC in triangle ABC with points A(-1,1), B(-4,5), and C(1,6). Learn about key geometric concepts, including midpoint calculation and the slope of lines, tailored for high school students.

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