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: Calculate the slope of the given line segment

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Given two points \((2,4)\) and \((4,- 2)\), then \(m=\frac{-2 - 4}{4 - 2}=\frac{-6}{2}=-3\).

Step2: Determine the slope of the perpendicular bisector

If two lines are perpendicular, the product of their slopes is \(-1\). Let \(m_1\) be the slope of the given line segment (\(m_1=-3\)) and \(m_2\) be the slope of the perpendicular bisector. Then \(-3\times m_2=-1\), so \(m_2=\frac{1}{3}\).

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

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

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

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

$$
LATEXBLOCK0
$$

Answer:

\(\boldsymbol{y=\frac{1}{3}x}\)