Answer:
The equation for the perpendicular bisector of the line segment will be:
[tex]y=-\frac{7}{2}x-13[/tex]
Step-by-step explanation:
Given the endpoints of the line segments
Determining the slope between (5,-4) and (-9, -8)
[tex]\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\left(x_1,\:y_1\right)=\left(5,\:-4\right),\:\left(x_2,\:y_2\right)=\left(-9,\:-8\right)[/tex]
[tex]m=\frac{-8-\left(-4\right)}{-9-5}[/tex]
[tex]m=\frac{2}{7}[/tex]
We know that a line perpendicular to another line contains a slope that is the negative reciprocal of the slope of the other line, such as:
slope = m = 2/7
Thus, the slope of the the new perpendicular line = – 1/m = (-1)/(2/7)= -7/2
Next, determining the mid-point between (5,-4) and (-9, -8)
[tex]\mathrm{Midpoint\:of\:}\left(x_1,\:y_1\right),\:\left(x_2,\:y_2\right):\quad \left(\frac{x_2+x_1}{2},\:\:\frac{y_2+y_1}{2}\right)[/tex]
[tex]\left(x_1,\:y_1\right)=\left(5,\:-4\right),\:\left(x_2,\:y_2\right)=\left(-9,\:-8\right)[/tex]
[tex]=\left(\frac{-9+5}{2},\:\frac{-8-4}{2}\right)[/tex]
Refine
[tex]=\left(-2,\:-6\right)[/tex]
We know that the point-slope form of equation of line is
[tex]y-y_1=m\left(x-x_1\right)[/tex]
where
substituting the slope of the perpendicular line -7/2 and the point (-2, -6)
[tex]y-y_1=m\left(x-x_1\right)[/tex]
[tex]y-\left(-6\right)=-\frac{7}{2}\left(x-\left(-2\right)\right)[/tex]
[tex]y+6=-\frac{7}{2}\left(x+2\right)[/tex]
Subtract 6 from both sides
[tex]y+6-6=-\frac{7}{2}\left(x+2\right)-6[/tex]
[tex]y=-\frac{7}{2}x-13[/tex]
Therefore, the equation for the perpendicular bisector of the line segment will be:
[tex]y=-\frac{7}{2}x-13[/tex]
