Algebra, Geometry, and Precalculus Help: Finding the Equation of the Perpendicular Bisector of a Line Segment

Somewhere along your math journey you will be expected to be able to find the equation of the perpendicular bisector of a line segment. When exactly you will encouter this particular math problem depends on your school and math program. Some students will cover this topic in Algebra, others in Geometry, and a few may not get it until Precalculus.

To solve these problems you will need to be familiar with the following:

Slope-intercept form of linear equations: y = mx + b

Point-slope form of linear equations: (y - y₁) = m(x - x₁)

Slope of a line going through two points P₁(x₁, y₁) and P₂(x₂, y₂) is (y₂-y₁)/(x₂-x₁)

You need to know that lines that are perpendicular to each other have slopes which are negative reciprocals of each other. (For example, if one line has a slope of -3, a line perpendicular to it would have a slope of 1/3.)

A "bisector" of a line segment is a line that goes through the midpoint of the segment. The midpoint of a line segment joining P₁(x₁, y₁) and P₂(x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2).

Sample Problem: How to Find the Equation of the Perpendicular Bisector of a Line Segment
Find the equation of the perpendicular bisector of the line segment joining (-4,-3) and (2,4) by using the point-slope form of the equation of a line.

First, we need to find the slope of the line segment: (4+3)/(2+4) = 7/6.

We know that the bisector is perpendicular, and so its slope is the negative reciprocal: -6/7.

Next we need to find the midpoint of the line segment: ((-4+2)/2, (-3+4)/2) = (-1, 1/2).

We plug in the midpoint and the slope of the bisector into the point-slope form:

(y - 1/2)=(-6/7)(x+1)

We simplify this further and we end up with:

y = (-6/7)x - (6/7) + (1/2)
y = (-6/7)x -(5/14)

And we have our final equation.

Blessings!

Source
Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen. Precalculus. Functions and Graphs. Fifth Edition