Understanding the Perpendicular Bisector of Line Segment AD and Its Equation

When analyzing line segments on a coordinate plane, determining the perpendicular bisector allows us to understand the geometric properties and relationships between points. Let's delve deeper into the equation of the perpendicular bisector of line segment AD, using the coordinates provided: A(-3, -3) and D(1, 3).

Interpreting the Problem

We are given the coordinates of points A and D, and we need to find the equation of the perpendicular bisector of the line segment AD. The perpendicular bisector of a line segment is a line that not only passes through the midpoint of the segment but also is perpendicular to it.

Step-by-Step Analysis

First, let's calculate the midpoint of the line segment AD using the midpoint formula:

Midpoint ( M ) of AD is given by:

[ M left( frac{x_A x_D}{2}, frac{y_A y_D}{2} right) ]

Substituting the given values:[ M left( frac{-3 1}{2}, frac{-3 3}{2} right) ]

This simplifies to:

[ M left( -1, 0 right) ]

The midpoint of AD is therefore ((-1, 0)).

Calculating the Slope

Next, we calculate the slope of the line segment AD. The slope (m) of a line passing through two points ((x_1, y_1)) and ((x_2, y_2)) is given by:

[ m frac{y_2 - y_1}{x_2 - x_1} ]

Substituting the coordinates of A and D:

[ m frac{3 - (-3)}{1 - (-3)} frac{6}{4} frac{3}{2} cdot frac{2}{2} frac{6}{4} frac{3}{2} cdot frac{1}{2} frac{3}{2} cdot frac{1}{2} 0 ]

This calculation shows that the slope of the line segment AD is actually 0, which indicates that it is a horizontal line.

Equation of the Perpendicular Bisector

A line that is perpendicular to a horizontal line is vertical. Therefore, the perpendicular bisector of AD, which passes through the midpoint ((-1, 0)), will be a vertical line.

The equation of a vertical line passing through a point with an x-coordinate (c) is:

[ x c ]

Since the midpoint of AD is ((-1, 0)), the equation of the perpendicular bisector is:

[ x -1 ]

Conclusion

In this analysis, we determined that the line segment AD is horizontal, and its perpendicular bisector is a vertical line passing through the midpoint ((-1, 0)). The equation of this perpendicular bisector is:

x -1

Understanding the properties of lines and their bisectors can be crucial in solving various geometric problems. This step-by-step approach not only clarifies the process but also reinforces the fundamental concepts of coordinate geometry.