Answer:
[tex]y=-2x-4[/tex]
Step-by-step explanation:
Perpendicular Bisector
The bisector of a segment defined by points (x1,y1) and (x2,y2) must pass by the midpoint of the segment.
The midpoint (xm,ym) is calculated as follows:
[tex]\displaystyle x_m=\frac{x_1+x_2}{2}[/tex]
[tex]\displaystyle y_m=\frac{y_1+y_2}{2}[/tex]
The endpoints of the segment are (-3,-8) and (5,-4), thus the midpoint M is:
[tex]\displaystyle x_m=\frac{-3+5}{2}=1[/tex]
[tex]\displaystyle y_m=\frac{-8-4}{2}=-6[/tex]
Midpoint: M(1,-6)
Let's find the slope of the given segment. The slope can be calculated with the formula:
[tex]\displaystyle m_1=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\displaystyle m_1=\frac{-4+8}{5+3}=\frac{1}{2}[/tex]
If the bisector is also perpendicular, its slope m2 and the slope of the segment m1 must comply:
[tex]m_1.m_2=-1[/tex]
Solving for m2:
[tex]\displaystyle m_2=-\frac{1}{m_1}=-\frac{1}{\frac{1}{2}}=-2[/tex]
Once we have the slope -2 and the point through which our line must pass (1,-6), we compute the equation in its point-slope form:
[tex]y-y_o=m(x-x_o)[/tex]
[tex]y-(-6)=-2(x-1)[/tex]
Operating
[tex]y+6=-2(x-1)[/tex]
[tex]y+6=-2x+2[/tex]
Rearranging
[tex]\boxed{y=-2x-4}[/tex]
