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Equations of Perpendicular Bisectors

In the realm of coordinate geometry, perpendicular bisectors play a crucial role in various mathematical applications. A perpendicular bisector is a line that passes through the midpoint of a line segment at a right angle (90 degrees). Understanding how to find the equation of a perpendicular bisector is essential for solving problems involving distances, symmetry, and geometric constructions.

Finding the Midpoint

Before we dive into the equations of perpendicular bisectors, it's important to review how to find the midpoint of a line segment. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint formula is:

$$ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) $$

Example

If we have a line segment with endpoints (2, 3) and (6, 7), the midpoint would be:

$$ \left(\frac{2 + 6}{2}, \frac{3 + 7}{2}\right) = (4, 5) $$

Perpendicular Slope

To find the equation of a perpendicular bisector, we need to understand the relationship between perpendicular lines. If two lines are perpendicular, their slopes are negative reciprocals of each other. This means that if the slope of one line is $m$, the slope of the line perpendicular to it is $-\frac{1}{m}$.

Note

The product of the slopes of perpendicular lines is always -1.

Steps to Find the Equation of a Perpendicular Bisector

Find the midpoint of the line segment.
Calculate the slope of the original line segment.
Determine the slope of the perpendicular bisector (negative reciprocal).
Use the point-slope form of a line to write the equation.

Let's break this down with an example:

Example

Find the equation of the perpendicular bisector of the line segment with endpoints (1, 2) and (5, 6).

Midpoint: $(\frac{1+5}{2}, \frac{2+6}{2}) = (3, 4)$
Slope of original line: $m = \frac{6-2}{5-1} = 1$
Slope of perpendicular bisector: $-\frac{1}{m} = -\frac{1}{1} = -1$
Using point-slope form: $y - y_1 = m(x - x_1)$ $y - 4 = -1(x - 3)$ $y - 4 = -x + 3$ $y = -x + 7$

Therefore, the equation of the perpendicular bisector is $y = -x + 7$.

Alternative Method: Using the General Form

Another approach to finding the equation of a perpendicular bisector is to use the general form of the line. If we have two points $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the perpendicular bisector can be written as:

$$ (x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2 $$

This equation represents all points that are equidistant from both endpoints of the line segment.

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Introduction to Perpendicular Bisectors

A perpendicular bisector is a line that divides a line segment into two equal parts at a 90-degree angle.
It passes through the midpoint of the segment and is perpendicular to it.

AnalogyThink of a perpendicular bisector as a perfectly balanced seesaw, where the pivot point is exactly in the middle, and the seesaw stands upright.

ExampleFor a line segment between points (0, 0) and (4, 0), the perpendicular bisector would be the vertical line $x = 2$ .

DefinitionPerpendicular BisectorA line that intersects a line segment at its midpoint and forms a 90-degree angle with it.

NoteThe perpendicular bisector is not the same as the median of a triangle, although they may coincide in some cases.

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

Internal Assessment (IA)3 subtopics

SL 3.5—Intersection of lines, equations of perpendicular bisectors Notes