To find the equation of the perpendicular bisector of the given line segment, follow these steps:
1. Identify the midpoint and the slope of the given line segment:
- Midpoint: [tex]\((-1, -2)\)[/tex]
- The slope of the given line segment: [tex]\(-4\)[/tex]
2. Determine the slope of the perpendicular bisector:
- Recall that the slope of the perpendicular bisector is the negative reciprocal of the slope of the given line segment.
- The slope of the given line segment is [tex]\(-4\)[/tex].
- Therefore, the perpendicular slope is [tex]\(\frac{1}{4}\)[/tex].
3. Use the point-slope form of the equation:
- The point-slope form is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line (which is the midpoint in this case), and [tex]\(m\)[/tex] is the slope.
- Plugging in the midpoint [tex]\((-1, -2)\)[/tex] and the perpendicular slope [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
y - (-2) = \frac{1}{4}(x - (-1))
\][/tex]
Simplify:
[tex]\[
y + 2 = \frac{1}{4}(x + 1)
\][/tex]
4. Solve for [tex]\(y\)[/tex] to put the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y + 2 = \frac{1}{4}x + \frac{1}{4}
\][/tex]
Subtract 2 from both sides:
[tex]\[
y = \frac{1}{4}x + \frac{1}{4} - 2
\][/tex]
Simplify the constant term:
[tex]\[
y = \frac{1}{4}x - \frac{8}{4} + \frac{1}{4} = \frac{1}{4}x - \frac{7}{4}
\][/tex]
5. Check the given choices to find the matching equation:
- [tex]\(y = -4x - 4\)[/tex]
- [tex]\(y = -4x - 6\)[/tex]
- [tex]\(y = \frac{1}{4}x - 4\)[/tex]
- [tex]\(y = \frac{1}{4}x - 6\)[/tex]
Considering the above constants and the slope of perpendicular bisector:
- Clearly, none of the given choices match [tex]\(\frac{1}{4}x - \frac{7}{4}\)[/tex].
Since none of the given options match the resulting slope-intercept form, the correct answer is none of these.
