To determine the equation of the perpendicular bisector of the given line segment, we need to follow these steps:
1. Identify the midpoint of the line segment:
The midpoint is given as [tex]\((3,1)\)[/tex].
2. Determine the slope of the given line:
The given line has the equation [tex]\(y = \frac{1}{3}x\)[/tex]. The slope ([tex]\(m\)[/tex]) of this line is [tex]\(\frac{1}{3}\)[/tex].
3. Find the slope of the perpendicular bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of the given line.
- The negative reciprocal of [tex]\(\frac{1}{3}\)[/tex] is [tex]\(-3\)[/tex].
Thus, the slope of the perpendicular bisector is [tex]\(-3\)[/tex].
4. Use the point-slope form to find the y-intercept:
The point-slope form of a line is given by [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
- We already determined that the slope [tex]\(m = -3\)[/tex].
- We know the midpoint [tex]\((3,1)\)[/tex] lies on this perpendicular bisector.
Using the equation [tex]\(y = mx + b\)[/tex] and plugging in the point [tex]\((3,1)\)[/tex]:
[tex]\[
1 = -3(3) + b
\][/tex]
[tex]\[
1 = -9 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = 1 + 9
\][/tex]
[tex]\[
b = 10
\][/tex]
5. Construct the equation in slope-intercept form:
Now, we have the slope [tex]\(m = -3\)[/tex] and the y-intercept [tex]\(b = 10\)[/tex]. Therefore, the equation of the perpendicular bisector in slope-intercept form is:
[tex]\[
y = -3x + 10
\][/tex]
After these steps, the resulting equation for the perpendicular bisector of the given line segment is:
[tex]\[
y = -3x + 10
\][/tex]
