Perpendicular Line Bisector

 Perpendicular line bisector will intersect the line at 90 degrees and divide the line in to two equal parts.

              The product of slope of the perpendicular line bisector and the line is -1.

              The intersection point of the line bisector and line is the midpoint of the line segment

 

Perpendicular line bisector

 

Steps to find the equation of the perpendicular line bisector:

 

 Step1:Find the slope of the line using the slope formula with two points

                Formula

                        Slope = `((y2-y1)/(x2-x1)) ` units

Step2:The slope of the perpendicular line bisector will be the negative reciprocal of the slope of the line

                       Slope of the perpendicular line bisector = - `(1/(slope ))`

Step3: Find the midpoint of the line segment using midpoint formula

Step4: Using the slope – point form formula, we can find the equation of the perpendicular line bisector.

 

 

 

Model problems for perpendicular line bisector:

 

 Ex: 1 Find the equation of the perpendicular line bisector of the line which passes through the points (2, 3) and (4, 5)?

Solution:            

                 Slope of the line= `((y2-y1)/(x2-x1))`

           Here

                             (x1,y1)= (2, 3)

                             (x2,y2)= (4, 5)

          Slope of the line = `((5-3)/ (4-2))`

                                        = `(2/2)`

                                       m  = 1

slope of the perpendicular line bisector = - `(1/ m)`

                                                                       = - `(1/1)`

 Slope of the perpendicular line bisector =-1    

Now the midpoint of the line = `((x1+x2)/2)` , `((y1+y2)/2)`

                                                   = `((2+4)/2)` , `((5+3)/2)`

                                                   = `(6/2)` , `(8/2)`

                                                   = (3, 4)

 Here the midpoint of the line segment is (3, 4)

Using the point and the slope of the line we can find the equation of the perpendicular line bisector.

        The equation is (y-y1) =m(x-x1)

                 Here the m is slope

                                 (x1,y1) = (3, 4)

                                    (y-4) = -1(x-3)

                                       y-4 =-x+3

                                      x+y =3+4

                                     x+y=7

         The equation of the perpendicular line bisector is x+y=7    

Ex: 2 Find the equation of the perpendicular line bisector of the line whose slope is 4 and the intersection of bisector with the line is (2, 4)?

 Sol:        the slope of the line, m = 4

               Slope of the perpendicular line bisector = - `(1/m)`

                                                                                = - `(1/4)`

               The equation is (y-y1) =m(x-x1)

                 Here the m is slope of line bisector

                                         (x1, y1) = (2, 4)

                                           (y-4) = - `(1/4)` (x-2)

                                           4(y-4) = - (x-2)

                                            4y-16 = -x+2

                                            x+4y = 2+16

                                             x+4y =18           

         The equation of the perpendicular line bisector is x+4y=18