To find the equation of the perpendicular bisector of the given line segment, we need to follow these steps:
1. Determine the Midpoint:
The midpoint of the line segment is given as [tex]\((-1, -2)\)[/tex].
2. Calculate the Slope of the Given Line:
The slope of the line [tex]\( y = -4x + b \)[/tex] can be directly read from the equation, which is [tex]\( -4 \)[/tex].
3. Find the Slope of the Perpendicular Bisector:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular bisector is:
[tex]\[
\text{Slope of perpendicular bisector} = \frac{1}{-(-4)} = \frac{1}{4}
\][/tex]
4. Formulate the Equation of the Perpendicular Bisector:
Using the point-slope form of the line equation [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the midpoint [tex]\((-1, -2)\)[/tex] and [tex]\(m\)[/tex] is the slope of the perpendicular bisector [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
y - (-2) = \frac{1}{4}(x - (-1))
\][/tex]
Simplifying, this becomes:
[tex]\[
y + 2 = \frac{1}{4}(x + 1)
\][/tex]
Expanding and isolating [tex]\(y\)[/tex]:
[tex]\[
y + 2 = \frac{1}{4}x + \frac{1}{4}
\][/tex]
[tex]\[
y = \frac{1}{4}x + \frac{1}{4} - 2
\][/tex]
[tex]\[
y = \frac{1}{4}x - 1.75
\][/tex]
Therefore, the equation of the perpendicular bisector in slope-intercept form is:
[tex]\[
y = \frac{1}{4}x - 1.75
\][/tex]
The step-by-step solution matches our determined numbers [tex]\( \frac{1}{4} \)[/tex] for the slope and [tex]\(-1.75\)[/tex] for the y-intercept, confirmed by the given result. Note that the exact equation form matches one of the provided options.
Looking at the provided choices, the correct answer is:
[tex]\[
\boxed{y = \frac{1}{4} x - 1.75}
\][/tex]
