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 with a midpoint at (-1, -2), we first need to know the coordinates of the endpoints of the segment. Assumed the line segment is between points (-5, -3) and (3, -1) based on the line you mentioned.

1. **Calculate the slope of the line segment:**

The slope (m) is calculated as follows:

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

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. **Determine the slope of the perpendicular bisector:**

The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment.

\[
\text{slope of perpendicular bisector} = -\frac{1}{m} = -\frac{1}{\frac{1}{4}} = -4
\]

3. **Use the midpoint as a point on the perpendicular bisector:**

We've found that the midpoint is \((-1, -2)\). Using the point-slope form of the line equation:

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

Plugging in \( m = -4 \) and the point (-1, -2):

\[
y - (-2) = -4(x - (-1))
\]

This simplifies to:

\[
y + 2 = -4(x + 1)
\]

Distributing the right side:

\[
y + 2 = -4x - 4
\]

Subtracting 2 from both sides gives:

\[
y = -4x - 6
\]

4. **Final Equation in Slope-Intercept Form:**

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

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