Perpendicular Bisector Finding the Equation

Summary

This educational video script explains the process of finding the equation of a line that is both perpendicular and bisecting a given line segment with endpoints (1, 2) and (5, 4). The midpoint is calculated, the slope of the original line is determined, and the opposite reciprocal slope is used for the perpendicular line. The slope-intercept form is applied to find the equation of the perpendicular bisector, which passes through the midpoint with a slope of -2, resulting in the equation y = -2x + 9. The script concludes with a brief graphing demonstration and an invitation to subscribe for more math tutorials.

Takeaways

• 📐 The video discusses finding a line that is perpendicular to a given line segment and bisects it.
• 📍 The first step is to find the midpoint of the segment using the formula (x1 + x2) / 2 and (y1 + y2) / 2.
• 📈 The midpoint in the example is calculated to be (3, 3) by averaging the x and y coordinates of the endpoints (1, 2) and (5, 4).
• 📉 To find the slope of the original line segment, the formula (y2 - y1) / (x2 - x1) is used, resulting in a slope of 1/2.
• 🔄 The slope of the perpendicular line is the negative reciprocal of the original slope, which is -2 in this case.
• 📝 The equation of the perpendicular line is derived using the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
• 🔍 The y-intercept (b) is found by substituting the midpoint coordinates into the equation and solving for b, which is 9 in the example.
• 📊 The final equation of the perpendicular bisector is y = -2x + 9.
• 🖋️ The video provides a brief demonstration of how to graph the line, showing it crosses the y-axis at (0, 9).
• 👨‍🏫 The presenter, Mario from Mario's Math Tutoring, encourages viewers to subscribe for more math tutoring videos.
• 🔗 The video concludes with an invitation to check out Mario's YouTube channel for further assistance.

Q & A

What is the midpoint of the line segment with endpoints (1, 2) and (5, 4)?

The midpoint is calculated using the formula ((x1 + x2)/2, (y1 + y2)/2). For the given endpoints, the midpoint is (3, 3).

How do you find the slope of a line segment?

The slope of a line segment is found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the endpoints.

What is the slope of the line segment connecting (1, 2) and (5, 4)?

The slope of the line segment is 1/2, calculated as (4 - 2) / (5 - 1) = 2 / 4 = 1/2.

Why do you need the opposite reciprocal to find the slope of a perpendicular line?

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. This is because perpendicular lines have slopes that multiply to -1.

What is the slope of the line perpendicular to one with a slope of 1/2?

The slope of the perpendicular line is the negative reciprocal of 1/2, which is -2.

How do you determine the equation of a line given a point and its slope?

The equation of a line can be determined using the point-slope form, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.

What is the equation of the line that is perpendicular to the segment (1, 2) to (5, 4) and passes through its midpoint (3, 3)?

Using the slope-intercept form y = mx + b, and knowing the slope is -2 and the line passes through (3, 3), the equation is y = -2x + 9.

How do you verify if a line bisects another line segment?

A line bisects another line segment if it passes through the midpoint of the segment and is perpendicular to it.

What is the y-intercept of the perpendicular bisector found in the script?

The y-intercept of the perpendicular bisector is 9, determined by substituting the midpoint (3, 3) into the equation y = -2x + b and solving for b.

Can you describe the process of graphing a line with a negative slope?

To graph a line with a negative slope, start at the y-intercept and for every unit increase in x, decrease the y-value by the slope's absolute value.

Outlines

00:00

📐 Finding the Midpoint and Slope of a Line Segment

This paragraph explains the process of finding the midpoint of a line segment with endpoints (1,2) and (5,4) by averaging the x-coordinates and y-coordinates, resulting in the midpoint at (3,3). It then describes calculating the slope of the original line segment using the slope formula, which yields a slope of 1/2. The speaker emphasizes the importance of finding the slope of the perpendicular line by taking the negative reciprocal, resulting in a slope of -2.

📈 Deriving the Equation of a Perpendicular Bisector

The paragraph continues by demonstrating how to derive the equation of a line that is perpendicular to the original line segment and passes through its midpoint (3,3). The slope of this new line is -2. The process involves using the slope-intercept form (y = mx + b) and solving for the y-intercept (b) by substituting the midpoint coordinates into the equation, leading to the final equation y = -2x + 9.

📚 Graphing the Perpendicular Bisector and Conclusion

The final part of the script includes a brief discussion on graphing the perpendicular bisector with the derived equation, noting that it crosses the y-axis at +9 and has a slope of -2, causing it to descend as it moves to the right. The speaker wraps up by encouraging viewers to subscribe for more math tutoring videos on their YouTube channel, Mario's Math Tutoring, and expresses eagerness to assist in future videos.

Keywords

💡endpoints

Endpoints refer to the two points that define the boundaries of a line segment. In the context of the video, the endpoints are the points labeled as '1' and '5' on the coordinate plane, which are used to determine the line segment's characteristics. The script discusses finding a perpendicular line that bisects the segment, starting with identifying these endpoints.

💡midpoint

The midpoint is the central point of a line segment, calculated as the average of the segment's x-coordinates and y-coordinates. The video script uses the midpoint formula to find the central point between the two endpoints, which is crucial for determining the location where the perpendicular bisector will intersect the original line segment.

💡perpendicular

Perpendicular lines are lines that intersect at a right angle, with a slope that is the negative reciprocal of the original line's slope. The script explains the process of finding a perpendicular line to a given segment, which involves calculating the slope of the original line and then finding its negative reciprocal.

💡slope

Slope is a measure of the steepness of a line, defined as the ratio of the vertical change to the horizontal change between two points on the line. The script calculates the slope of the original line segment and then finds the slope of the perpendicular line by taking the negative reciprocal.

💡slope formula

The slope formula is used to calculate the slope of a line given two points. It is expressed as (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. The script applies this formula to find the slope of the original line segment.

💡negative reciprocal

The negative reciprocal of a number is obtained by inverting the number and changing its sign. In the context of slopes, finding the negative reciprocal is essential for determining the slope of a line perpendicular to another. The script demonstrates this by taking the slope of the original line and finding its negative reciprocal to get the slope of the perpendicular line.

💡slope-intercept form

The slope-intercept form of a line is represented as y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The script uses this form to write the equation of the perpendicular line, once the slope and a point through which the line passes are known.

💡y-intercept

The y-intercept is the point where a line crosses the y-axis on a coordinate plane. It is represented by the 'b' value in the slope-intercept form of a line. The script finds the y-intercept by substituting the known x and y coordinates of the midpoint into the equation to solve for 'b'.

💡graphing

Graphing is the process of visually representing data or equations on a coordinate plane. In the script, graphing is used to illustrate the perpendicular line's path and its intersection with the y-axis, providing a visual aid to understanding the mathematical concepts discussed.

💡Mario's Math Tutoring

Mario's Math Tutoring appears to be the name of the channel or the tutor providing the educational content in the video. It is mentioned in the script as a source for more math tutoring videos, indicating that the channel offers additional resources for learning mathematics.

Highlights

Introduction to finding a line that is perpendicular and bisects a given segment.

Identifying the two endpoints of the segment as (1,2) and (5,4).

Explanation of the midpoint formula and its application.

Calculation of the midpoint coordinates as (3,3).

Marking the midpoint on the diagram for visual aid.

Using the slope formula to find the slope of the original line segment.

Determining the slope of the original line segment as 1/2.

Finding the slope of the perpendicular line by taking the opposite reciprocal.

Calculating the slope of the perpendicular line as -2.

Writing the equation of the line using the slope-intercept form.

Using the known point and slope to solve for the y-intercept.

Final equation of the perpendicular bisector: y = -2x + 9.

Brief demonstration of graphing the perpendicular bisector.

Crossing the y-axis at positive nine for the perpendicular bisector.

Invitation to subscribe for more educational content.

Promotion of the YouTube channel for additional math tutoring videos.

Anticipation of future video content and assistance.

Transcripts

so the first thing that we have here is

we have two endpoints of this segment

okay one two and five four and what we

want to do is we want to find the line

that's perpendicular so at a right angle

and it bisects it meaning it cuts it in

half so the first thing what we're going

to do is we're going to find that

midpoint that point halfway in between

the two endpoints so let's go ahead and

use our midpoint formula here notice

it's an average of the x-coordinates and

an average of the y-coordinates so let's

go ahead and write that down so we've

got the midpoint equals one plus five

divided by two and two plus four divided

by two so if we simplify that that's six

divided by two which is three 2 plus 4

is 6 divided by 2 which is also 3 so 3

comma 3 is our midpoint so let's go

ahead and mark that on our diagram so 3

3 right here okay so now what we're

going to do is we're going to find the

slope of this original line segment here

and so we're gonna use our slope formula

we're gonna take the y coordinate of

point 2 - the y coordinate of point 1

divided by the x coordinate of point 2

minus the x coordinate of point 1 so

let's go ahead and do that so we've got

4 minus 2 divided by 5 minus 1 okay so

we can see that's coming out to 2 over 4

which equals 1/2 so that means that this

is rising 1 running 2 rising one running

2 but if we want to find the slope of

the perpendicular line we need to take

the opposite reciprocal so opposite

means if we're gonna do the opposite of

positive 1/2 that's going to be negative

ok and then the reciprocal means we

could flip this over which is going to

be 2 over 1 or you could just say that's

equal to negative 2 okay so now what we

want to do is we want to write the

equation of this line but we've got the

point that the line goes through that's

3 3 and we've got the slope which is

negative 2 so let's go ahead and put

that together using our slope intercept

form of the line we're gonna use the y

equals MX plus B so let's write down

what we know so far so we have y equals

MX plus B the slope is negative 2

and the B value we don't actually know

that's the y-intercept that's where this

is crossing the y-axis but because we

have this point we can put the

x-coordinate in for x and the y

coordinate in for y and we can solve for

B so let's go ahead and do that so 3

equals negative 2 times 3 which is

negative 6 right plus B

add the six to both sides and you can

see that B is equal to nine so if we put

nine back in here we get y equals

negative 2x plus 9 now let's go ahead

and just see if we can graph it real

quick so through this point it's going

to have a slope of negative two so we're

going to go down two over one and if we

were to graph this it's going off the

board there but you can see it's going

to cross the y-axis way up here at

positive nine so I hope that helps you

understand how to work with the equation

of a perpendicular bisector subscribe to

the channel check out more math tutoring

videos on my youtube channel Mario's

math tutoring and I look forward to

helping you in the future videos I'll

talk to you soon