To find the equation of the perpendicular bisector of a line segment, we need to follow a series of steps. These steps include finding the midpoint of the segment, determining the slope of the original line segment, calculating the slope of the perpendicular bisector, and finally using the point-slope form of a line to write the equation of the perpendicular bisector in slope-intercept form.
Given:
- The midpoint of the line segment is [tex]\((3,1)\)[/tex].
- The equation of the line segment is [tex]\(y = \frac{1}{3}x\)[/tex].
### Step-by-Step Solution:
1. Find the slope of the original line segment.
The slope of the line segment is given by [tex]\(m = \frac{1}{3}\)[/tex].
2. Determine the slope of the perpendicular bisector.
The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line segment. Therefore, the slope of the perpendicular bisector [tex]\(m_\perp\)[/tex] is:
[tex]\[
m_\perp = -\frac{1}{\left(\frac{1}{3}\right)} = -3
\][/tex]
3. Use the midpoint to determine the y-intercept of the perpendicular bisector.
The midpoint is [tex]\((3, 1)\)[/tex]. We can now use the point-slope form of a line to find the y-intercept. The point-slope form is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plugging in the midpoint [tex]\((3,1)\)[/tex] and the slope [tex]\(-3\)[/tex]:
[tex]\[
y - 1 = -3(x - 3)
\][/tex]
Simplifying the equation:
[tex]\[
y - 1 = -3x + 9
\][/tex]
Adding 1 to both sides to solve for [tex]\(y\)[/tex] in slope-intercept form:
[tex]\[
y = -3x + 10
\][/tex]
4. Identify the correct equation from the given choices.
We need to find [tex]\(y = -3x + 10\)[/tex] among the provided options:
- [tex]\(y = \frac{1}{3}x\)[/tex]
- [tex]\(y = \frac{1}{3}x - 2\)[/tex]
- [tex]\(y = 3x\)[/tex]
- [tex]\(y = 3x - 8\)[/tex]
None of the options exactly match [tex]\(y = -3x + 10\)[/tex], however, the logic and calculations above confirm that [tex]\(y = -3x + 10\)[/tex] is indeed the equation of the perpendicular bisector.
Since none of the provided options match [tex]\(y = -3x + 10\)[/tex], it seems there may be a misunderstanding in the question or the options given. However, mathematically, the correct equation derived is:
[tex]\[ \boxed{y = -3x + 10} \][/tex]
