Answer:
y= -2x -8
Step-by-step explanation:
I will be writing the equation of the perpendicular bisector in the slope-intercept form which is y=mx +c, where m is the gradient and c is the y-intercept.
A perpendicular bisector is a line that cuts through the other line perpendicularly (at 90°) and into 2 equal parts (and thus passes through the midpoint of the line).
Let's find the gradient of the given line.
[tex]\boxed{gradient = \frac{y1 -y 2}{x1 - x2} }[/tex]
Gradient of given line
[tex] = \frac{1 - ( - 5)}{3 - ( - 9)} [/tex]
[tex] = \frac{1 + 5}{3 + 9} [/tex]
[tex] = \frac{6}{12} [/tex]
[tex] = \frac{1}{2} [/tex]
The product of the gradients of 2 perpendicular lines is -1.
(½)(gradient of perpendicular bisector)= -1
Gradient of perpendicular bisector
= -1 ÷(½)
= -1(2)
= -2
Substitute m= -2 into the equation:
y= -2x +c
To find the value of c, we need to substitute a pair of coordinates that the line passes through into the equation. Since the perpendicular bisector passes through the midpoint of the given line, let's find the coordinates of the midpoint.
[tex]\boxed{midpoint = ( \frac{x1 + x2}{2} , \frac{y1 + y2}{2}) }[/tex]
Midpoint of given line
[tex] = ( \frac{3 - 9}{2} , \frac{1 - 5}{2} )[/tex]
[tex] = ( \frac{ - 6}{2} , \frac{ - 4}{2} )[/tex]
[tex] = ( - 3 , - 2)[/tex]
Substituting (-3, -2) into the equation:
-2= -2(-3) +c
-2= 6 +c
c= -2 -6 (-6 on both sides)
c= -8
Thus, the equation of the perpendicular bisector is y= -2x -8.
