the given line segment has a midpoint at (3, 1). what is the equation, in slope - intercept form, of the…

Explanation:

Step1: Calculate the slope of the given line segment

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For the points $(2,4)$ and $(4, - 2)$, we have $m_1=\frac{-2 - 4}{4 - 2}=\frac{-6}{2}=-3$.

Step2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the perpendicular line be $m_2$. Then $m_1\times m_2=-1$. Since $m_1 = - 3$, we have $-3\times m_2=-1$, so $m_2=\frac{1}{3}$.

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 $y - y_0=m(x - x_0)$, where $(x_0,y_0)=(3,1)$ and $m = \frac{1}{3}$. So $y - 1=\frac{1}{3}(x - 3)$.

Step4: Convert to slope - intercept form

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

Answer:

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