To find the equation of the perpendicular bisector of a given line segment with a midpoint at [tex]\((3,1)\)[/tex] and given line [tex]\(y = \frac{1}{3} x\)[/tex]:
1. Determine the slope of the given line:
The given line is [tex]\(y = \frac{1}{3} x\)[/tex]. This is in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Therefore, the slope of the given line is [tex]\(\frac{1}{3}\)[/tex].
2. Find the slope of the perpendicular bisector:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. The negative reciprocal of [tex]\(\frac{1}{3}\)[/tex] is [tex]\(-3\)[/tex].
3. Use the point-slope form to write the equation of the perpendicular bisector:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1) = (3, 1)\)[/tex] is the midpoint, and [tex]\(m = -3\)[/tex] is the slope of the perpendicular bisector. Plug these values into the point-slope form:
[tex]\[
y - 1 = -3(x - 3)
\][/tex]
4. Simplify the equation to slope-intercept form:
Expand and simplify the equation:
[tex]\[
y - 1 = -3x + 9
\][/tex]
[tex]\[
y = -3x + 9 + 1
\][/tex]
[tex]\[
y = -3x + 10
\][/tex]
5. Compare with provided choices:
The resulting equation from our calculations is [tex]\(y = -3x + 10\)[/tex]. Now, let's compare this with the provided choices:
- [tex]\(y = \frac{1}{3} x\)[/tex]
- [tex]\(y = \frac{1}{3} x - 2\)[/tex]
- [tex]\(y = 3 x\)[/tex]
- [tex]\(y = 3 x - 8\)[/tex]
None of the provided choices match [tex]\(y = -3x + 10\)[/tex].
Therefore, there is no correct choice provided that matches the equation of the perpendicular bisector. The answer is that none of the choices are correct.
