Question

the given line segment has a midpoint at (3, 1). what is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment?
( \bigcirc ) ( y = \frac{1}{3}x )
( \bigcirc ) ( y = \frac{1}{3}x - 2 )
( \bigcirc ) ( y = 3x )
( \bigcirc ) ( y = 3x - 8 ) graph with points (2,4), (4,-2), midpoint (3,1) on a coordinate grid

Explanation:

Step1: Find slope of given segment

Points \((2,4)\) and \((4,-2)\). Slope \(m = \frac{-2 - 4}{4 - 2}=\frac{-6}{2}=-3\).

Step2: Find slope of perpendicular bisector

Perpendicular slope is negative reciprocal: \(m_{\perp}=\frac{1}{3}\).

Step3: Use point - slope form with midpoint \((3,1)\)

Point - slope: \(y - y_1 = m(x - x_1)\). Substitute \(m=\frac{1}{3}\), \(x_1 = 3\), \(y_1 = 1\):
\(y - 1=\frac{1}{3}(x - 3)\).

Step4: Simplify to slope - intercept form

\(y - 1=\frac{1}{3}x - 1\) → \(y=\frac{1}{3}x\).

Answer:

\(y = \frac{1}{3}x\) (Option: \(y=\frac{1}{3}x\))