Consider the line AB and its perpendicular bisector CD shown below.
The following steps would be useful to find the equation of the perpendicular bisector CD.
Step 1 :
Find the slope of AB.
Step 2 :
Find the midpoint of AB which is E.
Step 3 :
Since AB and CD are perpendicular,
slope of AB x slope of CD = -1
slope of CD = -1/slope of AB
Step 4 :
Perpendicular bisector CD is passing through the mid point of AB which is E. In slope intercept form equation of a line 'y = mx + b', using the slope of AD and point E, find the y-intercept 'b'.
Step 5 :
Write the equation of perpendicular bisector CD using the slope of CD, 'm' and y-intercept 'b'.
Example :
Find the equation of perpendicular bisector of the line joining the points A(-4, 2) and B(6, -4) in general form.
Solution :
Let E be the mid point and CD be the perpendicular bisector of AB.
Mid point of AB :
= E((x_{1} + x_{2})/2, (y_{1} + y_{2})/2)
Substitute (x_{1}, y_{1}) = A(-4, 2) and (x_{2}, y_{2}) = B(6, -4).
= E((-4 + 6)/2, (2 - 4)/2)
= E(2/2, -2/2)
= E(1, -1)
Slope of AB :
= (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (x_{1}, y_{1}) = A(-4, 2) and (x_{2}, y_{2}) = B(6, -4).
= (-4 - 2)/(6 + 4)
= -6/10
= -3/5
Slope of AD :
= -1/slope of AB
= -1/(-3/5)
= 5/3
Equation of the perpendicular bisector CD :
y = mx + b
Substitute m = 5/3.
y = (5/3)x + b ----(1)
Substitute (x, y) = E(1, -1).
-1 = (5/3)(1) + b
-1 = 5/3 + b
-1 - 5/3 = b
-8/3 = b
(1)----> y = (5/3)x - 8/3
Multiply each side by 5.
3y = 5x - 8
-5x + 3y + 8 = 0
5x - 3y - 8 = 0
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