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. $(-6,-3) ;(-8,-1)$

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.
$(-6,-3) ;(-8,-1)$

Key Concepts

Midpoint Formula

The midpoint formula is used to determine the exact center point of a line segment. It involves finding the average of the x-coordinates and the y-coordinates of the endpoints. This point represents the balance point of the segment and is essential in establishing the location of the perpendicular bisector.

Slope Formula

The slope formula calculates the steepness or incline of a line by comparing the vertical change (difference in y-coordinates) to the horizontal change (difference in x-coordinates) between two points. It is fundamental in analytic geometry for describing the direction and angle of a line.

Perpendicular Bisector

A perpendicular bisector of a line segment is a line that divides the segment into two equal parts at its midpoint and forms a right angle (90 degrees) with the segment. This concept is used in various geometric constructions and proofs, especially in problems involving circles, triangles, and symmetry.

Negative Reciprocal

When two lines are perpendicular in a Cartesian plane, their slopes are negative reciprocals of each other. This means that the product of the slopes is -1. Finding the negative reciprocal of the slope of a line gives the slope of the line that is perpendicular to it, which is crucial for constructing the equation of a perpendicular bisector.

Transcript

00:02 In this problem, we're given these two ordered pairs, and they're going to be the end points of a segment, and we need to find the line that is perpendicular to the segment.

00:12 And so if we just do a quick visual of what's going on, we have the point negative 22, negative 4, let's say that's over here, and we have the point negative 26.

00:22 Let's say that's up here.

00:24 And then we're looking at this segment, and we want the line that's perpendicular to it, that goes through the midpoint.

00:31 So let's say that's the midpoint.

00:33 We're looking for the equation of this line.

00:36 So we're going to need to know the midpoint, and keep in mind that my sketch is just a rough sketch, so that might not be to scale, that might not be totally accurate.

00:45 We need to know the midpoint, and we need to know the slope of our segment so that we can find the opposite reciprocal of it to get the slope of our line.

00:53 So let's start by finding the midpoint.

00:56 To find the midpoint, we average the x coordinates, negative 22 plus negative 22 over 2, and we average the y coordinates.

01:03 6 plus negative 4 over 2.

01:06 And that gives us negative 12 .1.

01:14 If you look back at the sketch, i could see how possibly that is the point negative 12 .1.

01:19 Mine is just a little low.

01:21 Okay, now we need the slope of the segment.

01:25 And the slope of the segment would be the change in y, negative 4 minus 6 over the change in x, negative 22 minus negative 2.

01:37 And that gives us negative 10 over negative 20, which is positive...