How to Write the Equation for the Perpendicular Bisector in Slope-Intercept Form
This tutorial provides a detailed, step-by-step guide on how to derive the equation for the perpendicular bisector of a line segment given its endpoints. By using the midpoint formula, slope calculations, and algebraic rearrangements, you'll learn how to convert the point-slope form of the equation to the slope-intercept form, a method that is often encountered in geometry and algebra. This article will equip you with the skills needed to tackle similar problems efficiently and accurately.
Introduction to the Problem
The problem we are addressing in this article involves finding the equation of the perpendicular bisector of a line segment joining two points, specifically M (-5, -4) and N (1, -2). A perpendicular bisector is a line that intersects the segment at its midpoint and forms a right angle with it. To solve this, we follow a structured approach that involves several key concepts: the midpoint formula, slope of a line, and the conversion of equation forms.
Step-by-Step Solution: Midpoint Calculation
The first step in solving the problem is to find the midpoint of the line segment joining points M and N. The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:
[(midpoint) left(frac{x1 x2}{2}, frac{y1 y2}{2}right)]For our specific points M (-5, -4) and N (1, -2):
[text{Midpoint} left(frac{-5 1}{2}, frac{-4 (-2)}{2}right) left(frac{-4}{2}, frac{-6}{2}right) (-2, -3)]Determine the Slope of the Line Segment
The next step is to calculate the slope of the line segment MN. The slope m of a line passing through two points (x1, y1) and (x2, y2) is given by:
[m frac{y2 - y1}{x2 - x1}]For our points M (-5, -4) and N (1, -2):
[m frac{-2 - (-4)}{1 - (-5)} frac{-2 4}{1 5} frac{2}{6} frac{1}{3}]Find the Equation of the Perpendicular Bisector
With the midpoint and the slope of the original line known, we now proceed to find the equation of the perpendicular bisector. The slope of the perpendicular bisector is the negative reciprocal of the slope of the original segment. Therefore, the slope of the perpendicular bisector m' is:
[m' -frac{1}{m} -frac{1}{frac{1}{3}} -3]The equation of the line in point-slope form is:
[frac{y - y1}{x - x1} m'(x - x1)]Substitute the midpoint coordinates (-2, -3) and the slope -3 into the point-slope form:
[frac{y - (-3)}{x - (-2)} -3(x - (-2))] [frac{y 3}{x 2} -3(x 2)]To convert this to slope-intercept form, (y mx b), we solve for (y):
[y 3 -3(x 2)] [y 3 -3x - 6] [y -3x - 6 - 3] [y -3x - 9]Alternative Method: Direct Perpendicular Bisector Equation
Alternatively, you can calculate the slope of the line segment MN and then write the equation of the perpendicular bisector directly in slope-intercept form:
[y -frac{1}{m}x b]Substitute the slope of the perpendicular bisector, -3, and the midpoint coordinates, (-2, -3), into the equation:
[-3 -3(-2) b] [-3 6 b] [b -3 - 6] [b -9]Hence, the equation of the perpendicular bisector is:
[y -3x - 9]Conclusion
By following these steps, you can find the equation of the perpendicular bisector in slope-intercept form. Understanding these methods not only strengthens your algebraic and geometric skills but also provides a versatile approach to similar problems. Whether you prefer the midpoint and slope approach or the direct slope-intercept method, both techniques offer valuable insights into the nature of lines and their relationships in the coordinate plane.
Key Terms
Slope-intercept form – A linear equation written in the standard form (y mx b), where (m) is the slope and (b) is the y-intercept. Perpendicular bisector – A line that intersects a segment at its midpoint and is perpendicular to it. Midpoint formula – A method for finding the midpoint of a line segment joining two points (A(x1, y1)) and (B(x2, y2)): (left(frac{x1 x2}{2}, frac{y1 y2}{2}right)).Additional Resources
For more related content, resources, and examples, feel free to visit our geometry and algebra section. Practice problems and detailed explanations are available to help deepen your understanding.