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?
options:
y = -4x - 4
y = -4x - 6
y = \frac{1}{4}x - 4
y = \frac{1}{4}x - 6
(graph shows a coordinate plane with points (-5, -3), (-1, -2), (3, -1) on a line segment)

Explanation:

Step1: Find slope of given segment

The two points on the given segment are \((-5, -3)\) and \((3, -1)\). The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). So, \(m=\frac{-1 - (-3)}{3 - (-5)}=\frac{2}{8}=\frac{1}{4}\).

Step2: Find slope of perpendicular bisector

The slope of a line perpendicular to a line with slope \(m\) is the negative reciprocal. So, the slope of the perpendicular bisector is \(-4\) (since the negative reciprocal of \(\frac{1}{4}\) is \(-4\)).

Step3: Use point - slope form with midpoint

The midpoint is \((-1, -2)\). The point - slope form is \(y - y_1=m(x - x_1)\). Substituting \(m=-4\), \(x_1=-1\), and \(y_1 = - 2\), we get \(y - (-2)=-4(x - (-1))\).
Simplify: \(y + 2=-4(x + 1)\).

Step4: Convert to slope - intercept form

Expand the right - hand side: \(y+2=-4x - 4\).
Subtract 2 from both sides: \(y=-4x - 6\).

Answer:

\(y = - 4x-6\) (corresponding to the option \(y=-4x - 6\))