Question

A line segment has endpoints at (4, –6) and (0, 2).

What is the slope of the given line segment?


What is the midpoint of the given line segment?


What is the slope of the perpendicular bisector of the given line segment?


What is the equation, in slope-intercept form, of the perpendicular bisector?

Answer 1

To solve your problem, let's break it down step by step.

### Step 1: Finding the Slope of the Line Segment

The slope \( m \) of a line segment that connects two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For the points \((4, -6)\) (let's denote it as \((x_1, y_1)\)) and \((0, 2)\) (let's denote it as \((x_2, y_2)\)), we have:

- \( x_1 = 4 \)
- \( y_1 = -6 \)
- \( x_2 = 0 \)
- \( y_2 = 2 \)

Now, substituting these values into the slope formula:

\[
m = \frac{2 - (-6)}{0 - 4} = \frac{2 + 6}{0 - 4} = \frac{8}{-4} = -2
\]

### Step 2: Finding 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 calculated using the formula:

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

Substituting the coordinates:

\[
M = \left(\frac{4 + 0}{2}, \frac{-6 + 2}{2}\right) = \left(\frac{4}{2}, \frac{-4}{2}\right) = (2, -2)
\]

### Step 3: Finding the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line segment. Since the slope of the line segment is \(-2\), the slope of the perpendicular bisector \( m_{\text{perpendicular}} \) is:

\[
m_{\text{perpendicular}} = -\frac{1}{-2} = \frac{1}{2}
\]

### Step 4: Finding the Equation of the Perpendicular Bisector

We can use the point-slope form of the equation of a line, which is:

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

where \((x_1, y_1)\) is the midpoint we found earlier \((2, -2)\), and \( m \) is the slope of the perpendicular bisector \(\frac{1}{2}\):

\[
y - (-2) = \frac{1}{2}(x - 2)
\]

This can be simplified to:

\[
y + 2 = \frac{1}{2}x - 1
\]

Subtracting \(2\) from both sides to find it in slope-intercept form:

\[
y = \frac{1}{2}x - 1 - 2 \quad \Rightarrow \quad y = \frac{1}{2}x - 3
\]

### Summary of Results

1. **Slope of the line segment:** \( -2 \)
2. **Midpoint of the line segment:** \( (2, -2) \)
3. **Slope of the perpendicular bisector:** \( \frac{1}{2} \)
4. **Equation of the perpendicular bisector (slope-intercept form):** \( y = \frac{1}{2}x - 3 \)