Respuesta :
Answer:
-3x+4y=4
Explanation:
Step 1: Find the slope of the line joining points (-1,-6) and (-7,2).
[tex]\begin{gathered} \text{Slope}=\frac{Change\text{ in y-axis}}{Change\text{ in x-axis}} \\ =\frac{-6-2}{-1-(-7)} \\ =\frac{-8}{-1+7} \\ =-\frac{8}{6} \\ m=-\frac{4}{3} \end{gathered}[/tex]Step 2: Find the midpoint of the line segment.
[tex]\begin{gathered} \text{Midpoint}=\mleft(\frac{-1+(-7)}{2},\frac{-6+2}{2}\mright) \\ =\mleft(\frac{-1-7}{2},\frac{-6+2}{2}\mright) \\ =\mleft(-\frac{8}{2},-\frac{4}{2}\mright) \\ =(-4,-2) \end{gathered}[/tex]Step 3: Find the slope of the perpendicular line.
Two lines are perpendicular if the product of their slopes = -1.
Let the slope of the perpendicular bisector =n
[tex]\begin{gathered} mn=-1 \\ -\frac{4}{3}n=-1 \\ n=\frac{3}{4} \end{gathered}[/tex]Step 4: Find the equation for the perpendicular bisector.
This is the equation of the line with a slope of 3/4 passing through (-4,-2).
[tex]\begin{gathered} y-y_1=n(x-x_1) \\ y-(-2)=\frac{3}{4}(x-(-4)) \\ y+2=\frac{3}{4}(x+4) \\ y=\frac{3}{4}(x+4)-2 \\ y=\frac{3}{4}x+3-2 \\ y=\frac{3}{4}x+1 \end{gathered}[/tex]The equation for the perpendicular bisector is:
[tex]\begin{gathered} y=\frac{3}{4}x+1 \\ y=\frac{3x+4}{4} \\ 4y=3x+4 \\ -3x+4y=4 \end{gathered}[/tex]