Question

the given line segment has a midpoint at $(-1,-2)$. what is the equation, in slope - intercept form, of the perpendicular bisector of the given line segment? $y=-4x - 4$ $y=-4x - 6$ $y=\frac{1}{4}x-4$ $y=\frac{1}{4}x - 6$

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 \((x_1,y_1)=(-5,-3)\) and \((x_2,y_2)=(3,-1)\).

$$m=\frac{-1-(-3)}{3 - (-5)}=\frac{-1 + 3}{3+5}=\frac{2}{8}=\frac{1}{4}$$

Step2: Find the slope of the perpendicular bisector

The slope of a perpendicular line to a line with slope \(m\) is \(m_{\perp}=-\frac{1}{m}\). Since \(m = \frac{1}{4}\), then \(m_{\perp}=-4\)

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

The perpendicular bisector passes through the mid - point \((x_0,y_0)=(-1,-2)\) and \(m=-4\).

$$y-(-2)=-4(x - (-1))$$

$$y + 2=-4(x + 1)$$

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

$$y+2=-4x-4$$

$$y=-4x-4 - 2$$

$$y=-4x-6$$

Answer:

\(y = - 4x-6\) (the second option)