How to Determine the Equation of the Perpendicular Bisector of a Line Segment

Introduction to Perpendicular Bisectors

In coordinate geometry, the equation of the perpendicular bisector of a line segment joining two points is a fundamental concept. The perpendicular bisector is a line that intersects the segment perpendicularly and divides it into two equal halves. This article explains how to find the equation of the perpendicular bisector for any given line segment.

Steps to Find the Equation of the Perpendicular Bisector

Step 1: Identify the Given Points

Consider two points, A(3,2) and B(7,6). The first step is to identify these coordinates.

Step 2: Calculate the Midpoint

The midpoint of the line segment is the point that divides the segment into two equal parts. The midpoint formula is given by:

[ M left( frac{x_1 x_0}{2}, frac{y_1 y_0}{2} right) ]

Substituting the coordinates of points A and B into the formula:

[ M left( frac{7 3}{2}, frac{6 2}{2} right) left( frac{10}{2}, frac{8}{2} right) (5, 4) ]

Step 3: Determine the Slope of the Line Segment

The slope of the line segment joining points A and B is calculated using the slope formula:

[ m frac{y_1 - y_0}{x_1 - x_0} frac{6 - 2}{7 - 3} 1 ]

The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment. Since the slope of the line segment is 1, the slope of the perpendicular bisector is -1.

Step 4: Write the Equation of the Perpendicular Bisector

The point-slope form of the equation of a line is given by:

[ y - y_1 m(x - x_1) ]

Using the midpoint (5,4) and the slope -1:

[ y - 4 -1(x - 5) ]

Simplifying this equation:

[ y - 4 -x 5 ] [ y -x 9 ]

General Formula for Perpendicular Bisectors

The general form of the equation of the perpendicular bisector of a line segment joining points (x_0, y_0) and (x_1, y_1) can be derived. The equation is:

[ 2x_1 - 2y_1 y x_1^2 - x_0^2 - y_1^2 y_0^2 ]

Substituting the values for points A(3,2) and B(7,6) into this formula:

[ 2(7) - 2(6)(-y) (7)^2 - (3)^2 - (6)^2 (2)^2 ]

This simplifies to:

[ 14 12y 49 - 9 - 36 4 ] [ 14 12y 8 ] [ 12y -6 ] [ y -frac{1}{2} cdot 6 ] [ y -0.5 cdot 6 ] [ y -3 cdot 2 ] [p]Thus, the equation of the perpendicular bisector is:

[ y -x 9 ]

Conclusion

The process of finding the equation of the perpendicular bisector of a line segment involves identifying the midpoint, determining the slope of the perpendicular bisector, and then applying the point-slope form of the line equation. Understanding these steps is crucial for solving geometric problems involving perpendicular bisectors.