EQUATION OF PERPENDICULAR BISECTOR

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((x1 + x2)/2, (y1 + y2)/2)

Substitute (x1, y1) = A(-4, 2) and (x2, y2) = B(6, -4).

= E((-4 + 6)/2, (2 - 4)/2)

= E(2/2, -2/2)

= E(1, -1)

Slope of AB : 

= (y2 - y1)/(x2 - x1)

Substitute (x1, y1) = A(-4, 2) and (x2, y2) = 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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