Question

The given line segment has a midpoint at (−1, −2).

On a coordinate plane, a line goes through (negative 5, negative 3), (negative 1, negative 2), and (3, negative 1).

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

y = −4x − 4
y = −4x − 6
y = One-fourthx – 4
y = One-fourthx – 6

Answer 1

To find the equation of the perpendicular bisector of the line segment that has a midpoint at (−1, −2) and passes through the points (−5, −3) and (3, −1), we can follow these steps:

1. **Find the slope of the original line segment.**
The slope \( m \) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \( (−5, −3) \) and \( (3, −1) \):
\[
m = \frac{-1 - (-3)}{3 - (-5)} = \frac{-1 + 3}{3 + 5} = \frac{2}{8} = \frac{1}{4}
\]

2. **Find the slope of the perpendicular bisector.**
The slope of the perpendicular bisector is the negative reciprocal of the original slope. Thus, if the slope of the original line is \( \frac{1}{4} \), the slope \( m_{\text{perpendicular}} \) is:
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{4}} = -4
\]

3. **Use the midpoint to write the equation of the perpendicular bisector.**
We have the slope of the perpendicular bisector as \( -4 \) and it passes through the midpoint \( (-1, -2) \). We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting the midpoint \( (-1, -2) \) and the slope \( -4 \):
\[
y - (-2) = -4(x - (-1))
\]
Simplifying that:
\[
y + 2 = -4(x + 1)
\]
\[
y + 2 = -4x - 4
\]
\[
y = -4x - 4 - 2
\]
\[
y = -4x - 6
\]

Therefore, the equation of the perpendicular bisector in slope-intercept form is:

\[
\boxed{y = -4x - 6}
\]