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? $y=\frac{1}{3}x$ $y=\frac{1}{3}x - 2$ $y = 3x$ $y = 3x - 8$

Explanation:

Step1: Find the slope of the given line segment

Use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(2,4)\) and \((x_2,y_2)=(4, - 2)\).

$$m=\frac{-2 - 4}{4 - 2}=\frac{-6}{2}=-3$$

Step2: Find the slope of the perpendicular bisector

The slope of a line perpendicular to a line with slope \(m\) is \(m_{\perp}=-\frac{1}{m}\). Since \(m = - 3\), then \(m_{\perp}=\frac{1}{3}\).

Step3: Use the point - slope form \(y - y_0=m(x - x_0)\)

The perpendicular bisector passes through the mid - point \((x_0,y_0)=(3,1)\) and \(m=\frac{1}{3}\).

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

Step4: Convert to slope - intercept form \(y=mx + b\)

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Answer:

\(y=\frac{1}{3}x\) (the first option)