To determine the equation of the perpendicular bisector of a line segment, follow these steps:
1. Identify the midpoint and the slope of the original line segment:
- The midpoint is given as [tex]\((-1, -2)\)[/tex].
- The original line segment is defined by the equations: [tex]\(y = -4x - 4\)[/tex] and [tex]\(y = -4x - 6\)[/tex].
- Notice that both lines share the same slope, which is [tex]\(-4\)[/tex].
2. Find the slope of the perpendicular bisector:
- The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
- The original slope is [tex]\(-4\)[/tex], so the perpendicular slope (m) is:
[tex]\[
m = -\frac{1}{-4} = \frac{1}{4}
\][/tex]
3. Use the point-slope form of the equation to find the equation of the perpendicular bisector:
- The point-slope form is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
- Substitute 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 the equation:
[tex]\[
y + 2 = \frac{1}{4}(x + 1)
\][/tex]
- Distribute [tex]\(\frac{1}{4}\)[/tex] on the right side:
[tex]\[
y + 2 = \frac{1}{4}x + \frac{1}{4}
\][/tex]
4. Convert the equation to slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Isolate [tex]\(y\)[/tex] on one side of the equation:
[tex]\[
y = \frac{1}{4}x + \frac{1}{4} - 2
\][/tex]
- Simplify the constants on the right side:
[tex]\[
y = \frac{1}{4}x + \frac{1}{4} - \frac{8}{4}
\][/tex]
[tex]\[
y = \frac{1}{4}x - \frac{7}{4}
\][/tex]
The equation of the perpendicular bisector in slope-intercept form is:
[tex]\[
y = \frac{1}{4} x - \frac{7}{4}
\][/tex]
