Bisectors of Triangles // GEOMETRY
Summary
TLDRIn this educational video, Gary LS from Gary Green explains the concept of 'concurrency' in triangles, focusing on the points where lines intersect. He introduces the 'circumcenter' as the point where the perpendicular bisectors of a triangle meet, equidistant from all vertices. The video also covers the 'incenter,' the point of concurrency for angle bisectors, which is always inside the triangle and equidistant from all sides. The content is presented in an engaging manner, with a touch of humor, to help viewers grasp these geometric concepts.
Takeaways
- 📌 The term 'concurrent' refers to three lines intersecting at a single point, known as the 'point of concurrency'.
- 📐 When drawing perpendicular bisectors in a triangle, they always intersect at a single point called the 'circumcenter'.
- 🔄 The circumcenter is equidistant from all the vertices of the triangle, regardless of the triangle's type.
- 🏞️ In an acute triangle, the circumcenter is located inside the triangle, while in an obtuse triangle, it is outside.
- 📍 For a right triangle, the circumcenter lies on the hypotenuse.
- 🌐 The circumcenter can be used to circumscribe a circle around the triangle, touching all vertices at equal distances.
- 🔶 The 'incenter' is the point of concurrency for the angle bisectors of a triangle and is always located inside the triangle.
- 📏 The incenter is equidistant from all the sides of the triangle, not the vertices, and this distance is measured perpendicularly to the sides.
- 🌀 A circle can be inscribed within the triangle that touches each side at the point equidistant from the incenter.
- 📝 The concept of 'concurrency' is central to understanding the properties of the circumcenter and incenter in the context of triangles.
- 🎨 The video script emphasizes the importance of visual representation and understanding geometric concepts through drawing and visualization.
Q & A
What does the term 'concurrent' refer to in the context of lines?
-In the context of lines, 'concurrent' refers to three or more lines that intersect at a single point, which is known as the point of concurrency.
What is the special name given to the point where the perpendicular bisectors of a triangle meet?
-The point where the perpendicular bisectors of a triangle meet is called the circumcenter.
What is the circumcenter's relationship to the vertices of a triangle?
-The circumcenter is equidistant from all the vertices of the triangle.
Where is the circumcenter located in relation to the triangle for different types of triangles?
-For an acute triangle, the circumcenter is inside the triangle. For an obtuse triangle, it is outside the triangle. For a right triangle, it lies on the hypotenuse.
What is meant by a circle being circumscribed around a triangle?
-A circle is said to be circumscribed around a triangle if it touches all the vertices of the triangle, with the circumcenter being the center of this circle.
What is the point of concurrency called when the angle bisectors of a triangle meet?
-When the angle bisectors of a triangle meet, the point of concurrency is called the incenter.
Is the incenter always inside the triangle?
-Yes, the incenter is always located inside the triangle.
What is unique about the incenter's distance to the sides of the triangle?
-The incenter is equidistant from all the sides of the triangle, meaning it is the same distance from each side when measured perpendicularly.
How does the incenter relate to the perpendicular bisectors of the triangle?
-The incenter is the point where the angle bisectors of the triangle meet, and it is equidistant from the perpendicular bisectors, not the sides.
What is the significance of the incenter in terms of balancing the triangle?
-The incenter is significant for balancing the triangle because it is the point where the triangle can be balanced on a pin, as it is the center of the inscribed circle that touches all three sides.
Outlines
📐 Understanding Triangle Bisectors and Points of Concurrency
The script introduces the concept of concurrent lines in triangles, where three lines intersect at a single point called the point of concurrency. It explains the perpendicular bisectors of a triangle, which intersect at the circumcenter, a point equidistant from all vertices. The video distinguishes the circumcenter's position based on the type of triangle: inside for acute, outside for obtuse, and on the hypotenuse for right triangles. It also emphasizes the circumcenter's role in circumscribing a circle around the triangle, touching all vertices at equal distances.
🔄 Exploring the Incenter and its Properties
This paragraph delves into the angle bisectors of a triangle, which intersect at the incenter, a point always located inside the triangle. The incenter is unique in that it is equidistant from all sides of the triangle, not the vertices. The script illustrates this by showing that perpendicular lines drawn from the incenter to each side are of equal length. The incenter is also the center of a circle inscribed within the triangle, touching each side at its midpoint. The summary highlights the importance of understanding the incenter's role in balancing and symmetry within the triangle.
Mindmap
Keywords
💡Concurrent
💡Point of Concurrency
💡Perpendicular Bisectors
💡Circumcenter
💡Equidistant
💡Acute Triangle
💡Obtuse Triangle
💡Right Triangle
💡Incenter
💡Angle Bisectors
Highlights
Gary LS introduces the concept of concurrent lines and the point of concurrency in a triangle.
Explanation of the term 'concurrent' and its significance in geometry.
Introduction to the concept of perpendicular bisectors and their role in triangle geometry.
The point where perpendicular bisectors meet is called the circumcenter of a triangle.
The circumcenter is equidistant from all vertices of the triangle.
Location of the circumcenter varies depending on the type of triangle: inside for acute, outside for obtuse, and on the hypotenuse for right triangles.
Circumcenter can be used to circumscribe a circle around a triangle, touching all vertices.
Transition to discussing angle bisectors and their significance in triangle geometry.
The point where angle bisectors meet is called the incenter of a triangle.
The incenter is always located inside the triangle and is equidistant from all sides.
The incenter is used to inscribe a circle within a triangle, touching all sides.
Difference between the circumcenter and incenter in terms of their distances from vertices and sides.
The incenter's unique property of being equidistant to all sides, requiring perpendicular lines to the sides.
Gary LS emphasizes the importance of understanding the concepts of circumcenter and incenter for geometric problem-solving.
Summary of the key differences between the circumcenter and incenter in a triangle.
Final review of the concepts discussed, reinforcing the understanding of triangle geometry related to concurrency.
Transcripts
[Music]
all right ladies and gentlemen today's
message is brought to you by Gary LS of
Gary green for all your suing needs all
right we're doing 5-2 bis sectors of
triangles all right bis sectors of
triangles this one's a doozy so take
some notes and Rewind it if you need to
stop it if you have to and just get
another teacher if I'm a Crum one all
right first word need to know is
concurrent so concurrent
that means that three
lines intersect that makes them
concurrent now that point right there
that little spot where they all hit is
called the point of
concurrency I'm be using that term quite
a bit so be good to pay attention to
that so we got concurrent and then point
of
concurrency all right now when we
talking about stuff with triangles a lot
but they all have points of concurrency
but but they all have different names
for that okay
on all right here we
go
R gosh that's an awful I've never do
that again all right let's try this
triangle right here say I want to draw
all three perpendicular bis sectors okay
all the perpendicular bis sectors
perpendicular means they're going to be
in the middle and it's going to be per
that's bisect and then that's
perpendicular so it cuts that in half
and that's perpendicular let's see
Cuts this in half right there I'd
say
perpendicular and then this one would be
right
here all right guess what that's
perpendicular as well they're always
going to hit when you do the
perpendicular bis
sectors for any triangle they're always
going to hit at the same spot that's
called their point of concurrency now
there's a special name for that okay
when the perpendicular bis sectors of a
triangle meet it is
called the circum Center if I'm saying
it right yep
circumcenter it's called the
circumcenter of the triangle now that's
only when the perpendicular bis sectors
all hit okay that's the point of
concurrency it's called the circumcenter
there's going to be a different name for
different stuff when we do other things
okay but that's what you need to know is
that's circumcenter
all right
now
um okay now this circum center right
here it is the exact same distance to
all the vertices like this
line this line and this line I didn't
draw that last one very
well those are all the exact same
distance
I know there's a lot of stuff going on
here but what you need to know is if you
have a triangle you do the perpendicular
bis sectors they all cross at one point
that's called the circum Center okay
that circumcenter is the same distance
to all the vertices to all the angles
okay goody goody gumdrops all right
let's move on to the next
one uh oh side note for circumcenter if
it's an acute triangle that point is
going to be inside the tri triangle if
it's an obtuse triangle it'll be outside
the triangle and if it's a right
triangle it's going to be on the
hypotenuse so some fun facts you can
take with you on your vacation okay
now now the circum Center we call it the
circum Center it is always circumscribed
inside the triangle circumscribed or uh
yeah circumscribes what it's called make
sure I don't say something wrong
everybody's going to make fun of me all
right what you do is that means that if
I made like a circle A Perfect Circle
that touched all these vertices which I
know I'm going to mess this
up all right let's pretend that that was
a perfect circle that circum Center it
would be in the exact middle of my
circle same distance to everything all
right I'm making it look more compli
because I thought it would look pretty
but it turns out it just looks like an
ugly triv Pursuit piece all right so
that's the circumcenter all right main
thing you need to know is the
circumcenter is where the perpendicular
bis sectors meet and it's equal distant
to all the
vertices goodness gracious I said that
like 19 times okay so we got perp
perpendicular bis sectors out of the way
now that was the three perpendicular bis
sectors next is the three angle bis
sectors okay
now I know you probably think that I
just edited that video because I went so
fast you probably couldn't even see me
it was like a blur but I didn't it was
on me took some n n classes in high
school don't worry about
it okay so the next one let's say we
have another
triangle okay that looks like a right
triangle but don't matter whatever I'll
do whatever I want all right first thing
we did was where the perpendicular bis
sectors meet now we're doing it where
the angle bis sectors meet
okay angle bis sectors cut this angle in
half looks something like that that one
looks something like that and this one
looks something like
this those are all angle bisector say I
cut that in half I cut that in half and
I cut that in
half okay right there where they all
meet their point of
concurrency is called the N
Center okay it's called the in center
and guess what it's in the center all
right like if you want to to balance
this full like like if I had a triangle
and I was trying to balance it on a pin
I would probably put it at the end
Center okay but that's just me I'm just
one man all right
now now uh in Center let me see what I'm
not telling you in Center is always
going to be inside the triangle always
you know circumcenter it was inside
outside or on the hypotenuse in Center
is always inside the triangle now also
you know how last time we put a triangle
uh we put a circle that went all the way
around it this time if we put a circle
inside of
it that touched every single one of
every single one of like the
perpendicular bis sectors it would make
it where that's the exact center it
doesn't look like it right there because
I'm a horrible artist I have no skills
of an artist and let's see there's one
more thing I need to tell you about in
center and I'll tell you right now all
right the in center of a tri triangle
you know how last time we did a
perpendicular bis sector thing and we
found the circumcenter and it was equal
distance equidistant to all the vertices
this one is equidistant to all the sides
if you go straight there make that
perpendicular make that
perpendicular make that perpendicular
that would be all the same length right
there those three right there okay and I
know it looks crappy cuz I have a lot of
stuff in here and I'm sorry please
forgive me
all
right but it's the same distance to all
those all right so perf I'm just going
keep reviewing and going back that way
you know first thing we did all the
perpendicular bis sectors their point of
concurrency is called the circum Center
okay it is equidistant to all the angles
the vertices this one angle bis sector
it uh is where you go from the angle B
sectors their point of concurrency is
always inside the triangle it is called
the in center and is equal distant to
all the sides but you have to go
straight to the sides remember what we
did last chapter talking about the
shortest distance to any side makes
perpendicular line yeah all right um I'm
out of breath so I think I'm just going
to end this one right here
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