Find an equation for the perpendicular bisector of the line segment whose endpoints are left parenthesis, 2, comma, minus, 1, right parenthesis(2,−1) and left parenthesis, minus, 6, comma, 3, right parenthesis(−6,3).

To find the equation of the perpendicular bisector of the line segment with endpoints \( (2, -1) \) and \( (-6, 3) \), we follow these steps:

### Step 1: Find the midpoint of the line segment
The midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Plugging in the given points \( (2, -1) \) and \( (-6, 3) \):
\[
M = \left( \frac{2 + (-6)}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{-4}{2}, \frac{2}{2} \right) = (-2, 1)
\]

### Step 2: Find the slope of the line segment
The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the given points:
\[
m = \frac{3 - (-1)}{-6 - 2} = \frac{4}{-8} = -\frac{1}{2}
\]

### Step 3: Find the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the original slope. Therefore, if the original slope is \( -\frac{1}{2} \), the perpendicular slope \( m_{\perp} \) is:
\[
m_{\perp} = 2
\]

### Step 4: Use the point-slope form to find the equation of the perpendicular bisector
The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. Using the midpoint \( (-2, 1) \) and the perpendicular slope \( 2 \), we get:
\[
y - 1 = 2(x + 2)
\]

### Step 5: Simplify the equation
Expanding the equation:
\[
y - 1 = 2x + 4
\]
\[
y = 2x + 5
\]

Thus, the equation of the perpendicular bisector is:
\[
y = 2x + 5
\]

Would you like more details or have any questions about the process?

Here are 5 related questions for further practice:
1. How do you find the equation of a line passing through two given points?
2. What is the significance of the negative reciprocal in finding the perpendicular slope?
3. How would the solution change if one endpoint was at the origin?
4. Can you find the distance between the original endpoints?
5. How do you determine if two lines are parallel?

**Tip:** Always double-check calculations for slopes and midpoint, as they are crucial for accuracy in line equations.

Find an equation for the perpendicular bisector of the line segment whose endpoints are (2, −1) and (−6, 3).

Learn how to find the equation of the perpendicular bisector of a line segment with endpoints at (2, -1) and (-6, 3). This solution includes step-by-step calculations using midpoint and slope formulas, and covers key geometry concepts for Grades 9-11 students.

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