Write an equation in point-slope form for the perpendicular bisector of the segment with the giv

Write an equation in point-slope form for the perpendicular...

Key Concepts

Point-Slope Form

The point-slope form of a linear equation, written as y - y1 = m(x - x1), provides a method to define a line when a point on the line and its slope are known. It is particularly useful in forming the equation of a perpendicular bisector once the appropriate point and slope have been determined.

Perpendicular Lines

When two lines are perpendicular, the product of their slopes is -1. This means that the slope of the perpendicular bisector is the negative reciprocal of the slope of the original line, allowing us to determine the correct orientation for the bisector.

Midpoint Formula

The midpoint formula is used to find the center point of a line segment by averaging the x-coordinates and the y-coordinates of its endpoints. This point is crucial for constructing a perpendicular bisector because it lies exactly halfway along the segment.

Slope Calculation

Calculating the slope involves determining the ratio of the difference of the y-coordinates to the difference of the x-coordinates between two points. This concept is essential for identifying the direction and steepness of a line and is given by the formula m = (y2 - y1) / (x2 - x1).

Transcript

00:01 For problem 21, we are finding the equation of the perpendicular bisector for the segment mn.

00:08 So to help me visually, i'm going to go ahead and graph the segment mn so we can see what that looks like.

00:14 So now that we have our segment, we can kind of visualize the perpendicular bisector.

00:19 Now remember, perpendicular bisectors cut through the segment perpendicularly, which means they make a right angle, and they bisect, which means they cut the segment in equal parts.

00:35 So they want us to write our equation in point slope form.

00:38 So for point slope form, we need two things.

00:40 We need a point on the line and the slope of the line.

00:45 So the point we're going to use is this point here, because that point is the midpoint of the segment mn, because it cuts it in half.

00:54 So when we find the midpoint, then we need to find the slope.

01:02 Well, perpendicular lines have perpendicular slopes.

01:06 So if we can find the slope of mn, then we can change it to its perpendicular counterpart to find the slope of our line.

01:15 So we've got some things to do.

01:17 First, let's use the midpoint formula to find the midpoint of segment mn.

01:22 So i have written out the midpoint formula, and i went ahead and labeled our points as x1, y, 1, x2, y2, just to make it easier for us to plug it into the formula.

01:32 Now when we finish this formula, it's going to come out to our exact point, which is very nice.

01:37 Remember when labeling that the first point of the statement is always your x1y1.

01:42 So x1 is negative 3 plus x2 is 7 divided by 2.

01:51 Y1 is negative 1 plus y2 is negative 5 divided by 2.

01:58 Now you can simplify this many ways.

02:00 I just go ahead and do it all together.

02:02 You can just do the x's, then just do the y's.

02:05 Either way, we're going to come out with the same answer.

02:08 So, negative 3 plus 7 is 4, and we're going to divide that by 2.

02:14 Negative 1 plus negative 5 is negative 6, and we're going to divide that by 2.

02:19 Now we simplify by dividing by 2, 4 divided by 2 is 2, and negative 6 divided by 2 is negative 3.

02:27 So the midpoint of our segment is 2 negative 3, which lands about right here.

02:35 I don't have a perfect scale here, but we can see that 2 negative 3 is here.

02:42 So that point cuts this segment into two equal parts.

02:47 So that's our point for our point slope form...

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