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.