Perpendicular Bisector Finding the Equation
Summary
TLDRThis 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
📐 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.
Mindmap
Keywords
💡endpoints
💡midpoint
💡perpendicular
💡slope
💡slope formula
💡negative reciprocal
💡slope-intercept form
💡y-intercept
💡graphing
💡Mario's Math Tutoring
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
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