The form of the equation of the line is
y = m x + b, where
m is the slope
b is the y-intercept
The rule of the slope is
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
(x1, y1) and (x2, y2) are two points on the line
To find the equation of the perpendicular bisector to the line whose endpoints are (-8, -3) and (2, 3), we must find the slope of this line
x1 = -8 and y1 = -3
x2 = 2 and y2 = 3
Substitute them in the rule above
[tex]m=\frac{3--3}{2--8}=\frac{3+3}{2+8}=\frac{6}{10}=\frac{3}{5}[/tex]
To find the slope of the perpendicular line reciprocal the fraction and change its sign
The slope of the perpendicular line is
[tex]-\frac{5}{3}[/tex]
Substitute it in the form of the equation
[tex]y=-\frac{5}{3}x+b[/tex]
To find b we must have a point on the line
Since the line is the bisector of the given line, then it passes through its mid-point, then we need to find the mid-point of the given line
[tex]M=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]
We will use the endpoints above
[tex]\begin{gathered} M=(\frac{-8+2}{2},\frac{-3+3}{2}) \\ M=(-\frac{6}{2},\frac{0}{2}) \\ M=(-3,0) \end{gathered}[/tex]
The mid-point is (-3, 0)
We will use it to find b
Substitute x by -3 and y by 0 in the equation to find b
[tex]\begin{gathered} 0=-\frac{5}{3}(-3)+b \\ 0=5+b \end{gathered}[/tex]
Subtract 5 from both sides
-5 = b + 5 - 5
-5 = b
Substitute it in the equation
[tex]\begin{gathered} y=-\frac{5}{3}x+(-5) \\ y=-\frac{5}{3}x-5 \end{gathered}[/tex]
The equation of the perpendicular bisector is
[tex]y=-\frac{5}{3}x-5[/tex]
