To determine the equation of the perpendicular bisector of a line segment given its midpoint, let's follow these steps.
1. Identify the midpoint: The midpoint of the line segment is [tex]\((-1,-2)\)[/tex].
2. Determine the slope of the original line segment: Since we aren't given the original line segment directly, we will infer the properties we need. The original line segment isn't directly provided, but we know that the perpendicular bisector must be perpendicular to the original line passing through [tex]\((-1,-2)\)[/tex].
3. Compute the slope of the perpendicular bisector: The slopes of the perpendicular lines are negative reciprocals of each other. Let's find which original line is perpendicular to our bisector.
4. Examine potential lines for perpendicularity:
- First line: [tex]\(y = \frac{1}{4}x - 4\)[/tex]
- Second line: [tex]\(y = \frac{1}{4}x - 6\)[/tex]
- Third line: [tex]\(y = -4x - 6\)[/tex]
Both the first and second lines have a slope of [tex]\(\frac{1}{4}\)[/tex]. The negative reciprocal of [tex]\(\frac{1}{4}\)[/tex] is [tex]\(-4\)[/tex].
5. Find the equation of the perpendicular bisector: The slope of the perpendicular bisector derived from the properties should be [tex]\(-4\)[/tex], matching the third given line. The perpendicular line is expected to have the slope [tex]\(-4\)[/tex] and should pass through the midpoint [tex]\((-1,-2)\)[/tex].
6. Verify correctness:
- Equation of the third line: [tex]\(y = -4x - 6\)[/tex].
- Now, check if it passes through [tex]\((-1,-2)\)[/tex]:
[tex]\[ y = -4(-1) - 6 = 4 - 6 = -2 \][/tex]
The equation holds true for our point.
Since all steps are satisfied by the third line, the correct equation for the perpendicular bisector passing through the midpoint [tex]\((-1,-2)\)[/tex] and having the proper characteristics is:
[tex]\[ y = -4x - 6 \][/tex]
Thus, the equation of the perpendicular bisector is:
[tex]\[ y = -4x - 6 \][/tex]
And this corresponds to the option:
[tex]\[ \boxed{3} \][/tex]
