Perpendicular Bisector of a Line Segment Definition

Definition

Perpendicular bisector is a line or a ray which cuts another line segment into two equal parts at 90 degree. Bisector is simply a line or a ray which cuts another line segment into two equal parts. In the below image, AB is the perpendicular bisector of the line segment PQ and F is the midpoint of the line segment PQ.


perpendicular-bisector

Perpendicular Bisector of a Line Segment

Perpendicular Bisector Equation

Lets find it with points P(5,7), Q(6,6). Consider the co-ordinates of the points P and Q to be x1,y1 and x2,y2 respectively. We need to calculate the midpoints of the line PQ, which is F, and the slope to find the equation of the perpendicular bisector.

Step 1

Lets calculate the midpoint of the line which is the average of the x and y co-ordinates. Midpoint of a line = x1+x2/2, y1+y2/2 Midpoint of PQ = 5+6/2, 7+6/2 = (11/2, 13/2)

Step 2

Next, we need to find the slope of the line PQ using the formula y2-y1/x2-x1. Kindly note that the slope is represented by the letter 'm'. Slope of PQ (m) = 6-7/6-5 = -1.

Step 3

Now, lets calculate the slope of the perpendicular bisector (AB) of the line PQ. The slope of the perpendicular bisector = -1/slope of the line. Therefore for AB= -1/-1 = 1

Step 4

Once we find the slope as above, we can find the equation with the slope and the midpoints. Lets find the equation of the AB with midpoints (11/2,13/2) and the slope 1. Formula to find the equation y-y1 = m(x-x1) y-13/2 = 1(x-11/2) By solving the above, we get the equation -x + y = 1. This is the perpendicular bisector equation (AB) of the line PQ.

This tutorial helps to learn the definition and the calculation of perpendicular bisector of a line segment with example.

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