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?
options:
( y = \frac{1}{3}x )
( y = \frac{1}{3}x - 2 )
( y = 3x )
( y = 3x - 8 )
(graph shows points (2,4) and (4,-2) with midpoint (3,1) on a coordinate grid.)

Explanation:

Step1: Find slope of given segment

Given 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: \(y - y_1 = m(x - x_1)\), so \(y - 1 = \frac{1}{3}(x - 3)\).

Step4: Simplify to slope - intercept form

\(y - 1 = \frac{1}{3}x - 1\), add 1: \(y = \frac{1}{3}x\). Wait, no—wait, midpoint is \((3,1)\), let's re - check. Wait, when we expand \(y - 1=\frac{1}{3}(x - 3)\), it's \(y - 1=\frac{1}{3}x - 1\), then \(y=\frac{1}{3}x\)? But wait, let's check the options. Wait, maybe I made a mistake. Wait, the given segment has points \((2,4)\) and \((4, - 2)\), midpoint \((3,1)\) is correct. Slope of segment: \(\frac{-2 - 4}{4 - 2}=\frac{-6}{2}=-3\), so perpendicular slope is \(\frac{1}{3}\). Now, using point - slope with \((3,1)\): \(y - 1=\frac{1}{3}(x - 3)\). Distribute: \(y - 1=\frac{1}{3}x - 1\). Add 1: \(y=\frac{1}{3}x\). But wait, the options have \(y = \frac{1}{3}x\) as option A. Wait, but let's check again. Wait, maybe the midpoint is correct, and the calculation is right. So the equation is \(y=\frac{1}{3}x\). Wait, but let's check the graph. The midpoint is \((3,1)\), and the perpendicular bisector should pass through \((3,1)\). Let's plug \(x = 3\) into \(y=\frac{1}{3}x\), we get \(y = 1\), which matches the midpoint. So that's correct.

Answer:

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