Example: Find an equation of the perpendicular bisector of...

Example: Find an equation of the perpendicular bisector of the line seg. ment whose endpoints are $(2,6)$ and $(0,-2)$. Graph can't copy Solution: A perpendicular bisector is a line that contains the midpoint of the given segment and is perpendicular to the segment. Step 1. The midpoint of the segment with endpoints $(2,6)$ and $(0,-2)$ is $(1,2)$. Step 2. The slope of the segment containing points $(2,6)$ and $(0,-2)$ is 4 . Step 3. A line perpendicular to this line segment will have slope of $-\frac{1}{4}$. Step 4. The equation of the line through the midpoint $(1,2)$ with a slope of $-\frac{1}{4}$ will be the equation of the perpendicular bisector. This equation in standard form is $x+4 y=9$. Find an equation of the perpendicular bisector of the line segment whose endpoints are given. See the previous example. $(-6,8) ;(-4,-2)$

Example:
Find an equation of the perpendicular bisector of the line seg. ment whose endpoints are $(2,6)$ and $(0,-2)$.
Graph can't copy
Solution:
A perpendicular bisector is a line that contains the midpoint of the given segment and is perpendicular to the segment.
Step 1. The midpoint of the segment with endpoints $(2,6)$ and $(0,-2)$ is $(1,2)$.
Step 2. The slope of the segment containing points $(2,6)$ and $(0,-2)$ is 4 .
Step 3. A line perpendicular to this line segment will have slope of $-\frac{1}{4}$.
Step 4. The equation of the line through the midpoint $(1,2)$ with a slope of $-\frac{1}{4}$ will be the equation of the perpendicular bisector. This equation in standard form is $x+4 y=9$.

Find an equation of the perpendicular bisector of the line segment whose endpoints are given. See the previous example.
$(-6,8) ;(-4,-2)$

Key Concepts

Point-Slope Form of a Line

The point-slope form is a way of expressing the equation of a line when a point on the line and its slope are known. It is written as y - y? = m(x - x?), where (x?, y?) is the point and m is the slope. This form is especially useful for deriving the equation of a line in problems where only a single point and the slope are provided.

Negative Reciprocal Relationship

When 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 will be -1/m. This property is used to find the slope of any line perpendicular to a given line, which is particularly useful in problems involving perpendicular bisectors.

Perpendicular Bisector

A perpendicular bisector is a line that divides a given line segment into two equal parts while intersecting it at a right angle. It is an important concept in geometry because it is used in constructions, proofs, and solving problems that involve symmetry and distance relations.

Midpoint

The midpoint of a line segment is the point that is exactly halfway between the two endpoints. It is calculated by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints. This concept is crucial when finding the center of a segment, especially when constructing bisectors.

Slope

The slope of a line is a measure of its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between any two distinct points on the line. Understanding slope is essential for analyzing linear relationships and determining the orientation of lines in the plane.

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