Understanding the Perpendicular Bisector of a Line Segment

Finding the equation of the perpendicular bisector of a line segment joining two points is a fundamental concept in geometry, often useful in various applications such as computer graphics, engineering, and real-world construction. This article will guide you through the process using different methods. We will explore the method using the mid-point and the slope, as well as a method involving the general equation of a circle.

Method 1: Using the Midpoint and Slope

To find the equation of the perpendicular bisector of a line segment joining points A(2, 3) and B(6, -5), we start by calculating the mid-point of the segment AB. The mid-point M is given by:

[ M left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right) ] For points A(2, 3) and B(6, -5), the mid-point is: [ M left( frac{2 6}{2}, frac{3 (-5)}{2} right) (4, -1) ] Next, we calculate the slope of the line segment AB:

[ text{slope of } AB frac{y_2 - y_1}{x_2 - x_1} frac{-5 - 3}{6 - 2} frac{-8}{4} -2 ] The slope of the perpendicular bisector will be the negative reciprocal of the slope of AB:

[ text{slope of perpendicular bisector} frac{1}{2} ] Using the point-slope form of the equation of a line, we can write the equation of the perpendicular bisector passing through the mid-point M(4, -1) and having a slope of ( frac{1}{2} ): [ y - y_1 m(x - x_1) ] Substituting the values, we get:

[ y - (-1) frac{1}{2}(x - 4) ] [ y 1 frac{1}{2}x - 2 ] [ y frac{1}{2}x - 3 ] Thus, the equation of the perpendicular bisector is: [ y frac{1}{2}x - 3 ]

Method 2: Using the General Equation of a Circle

Another method involves using the general equation of a circle. If the line segment AB is the diameter of the circle, the equation of the circle can be derived as:

[ x^2 y^2 - 2ax - 2by a^2 b^2 - R^2 0 ] For point A(2, 3), we have: [ 2^2 3^2 - 2a(2) - 2b(3) a^2 b^2 - R^2 0 ] For point B(6, -5), we have: [ 6^2 (-5)^2 - 2a(6) - 2b(-5) a^2 b^2 - R^2 0 ] Using the substitutions and simplifying, we can derive the equation of the circle and then the perpendicular bisector. However, the result will be the same, and the perpendicular bisector is given as: [ 2y - x 6 0 ]

Summary of Key Concepts

The key concepts used in these methods include:

The mid-point formula: Calculated as (( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} )) The slope of the line segment: Calculated as (frac{y_2 - y_1}{x_2 - x_1}) The slope of the perpendicular bisector: The negative reciprocal of the slope of the line segment. The point-slope form of the equation of a line: ( y - y_1 m(x - x_1) ) By understanding these basic geometric principles, you can effortlessly determine the equation of the perpendicular bisector of any line segment that joins two points.

Conclusion

Knowing how to find the equation of the perpendicular bisector of a line segment is a valuable skill in many fields. Whether you are dealing with engineering, architectural design, or simply solving a geometry problem, being able to use these methods accurately will save you time and effort. By following the steps outlined in this article, you can confidently find the perpendicular bisector of any given line segment.