Question

the given line segment has a mid - point at (3, 1). what is the equation, in slope - intercept form, of the perpendicular bisector of the given line segment? y = -\frac{1}{3}x y = -\frac{1}{3}x - 2 y = 3x y = 3x - 8

Explanation:

Step1: Find the slope of the given line segment

The two - points on the line segment are $(2,4)$ and $(4, - 2)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
$m=\frac{-2 - 4}{4 - 2}=\frac{-6}{2}=-3$

Step2: Find the slope of the perpendicular bisector

The slope of a line perpendicular to a line with slope $m$ is the negative reciprocal. If the slope of the given line is $m=-3$, then the slope of the perpendicular bisector $m_{perp}=\frac{1}{3}$.

Step3: Use the point - slope form to find the equation of the perpendicular bisector

The perpendicular bisector passes through the mid - point $(3,1)$. The point - slope form of a line is $y - y_1=m(x - x_1)$. Here, $x_1 = 3$, $y_1 = 1$ and $m=\frac{1}{3}$.
$y - 1=\frac{1}{3}(x - 3)$

Step4: Convert to slope - intercept form

Expand the right - hand side: $y - 1=\frac{1}{3}x-1$.
Add 1 to both sides: $y=\frac{1}{3}x$.

Answer:

$y=\frac{1}{3}x$