Answer:
[tex]y=\frac{1}{3}x[/tex]
Step-by-step explanation:
We have been given an image of a line segment and we are asked to find the equation of perpendicular bisector of our given line segment in slope-intercept form.
Since we know that the slope of perpendicular line to a given line is negative reciprocal of the slope of the given line.
Let us find the slope of our given line using slope formula.
[tex]\text{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex], where,
[tex]y_2-y_1[/tex] = Difference between two y-coordinates,
[tex]x_2-x_1[/tex] = Difference between two x-coordinates of same y-coordinates.
Upon substituting the coordinates of points (2,4) and (4,-2) in slope formula we will get,
[tex]\text{Slope}=\frac{-2-4}{4-2}[/tex]
[tex]\text{Slope}=\frac{-6}{2}[/tex]
[tex]\text{Slope}=-3[/tex]
Now we will find negative reciprocal of [tex]-3[/tex] to get the slope of perpendicular line.
[tex]\text{Negative reciprocal of }-3=-(-\frac{1}{3})=\frac{1}{3}[/tex]
Since point (3,1) lies on the perpendicular line, so we will substitute coordinates of point (3,1) in slope-intercept form of equation [tex](y=mx+b)[/tex].
[tex]1=\frac{1}{3}\times 3+b[/tex]
[tex]1=1+b[/tex]
[tex]1-1=1-1+b[/tex]
[tex]b=0[/tex]
Therefore, the equation of perpendicular line will be [tex]y=\frac{1}{3}x+0=\frac{1}{3}x[/tex] and option A is the correct choice.
