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The given line segment has a midpoint at [tex][tex]$(-1, -2)$[/tex][/tex].

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

A. [tex]y = -4x - 4[/tex]
B. [tex]y = -4x - 6[/tex]
C. [tex]y = \frac{1}{4}x - 4[/tex]
D. [tex]y = \frac{1}{4}x - 6[/tex]


Sagot :

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]