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Current time:0:00Total duration:12:58

CCSS Math: 4.G.A.1

What I want to do in this
video is give an introduction to the language or
some of the characters that we use when we
talk about geometry. And I guess the
best place to start is even think about
what geometry means, as you might recognize the
first part of geometry right over here. You have the root word
geo-- the same word that you see in things
like geography and geology. And this refers to the Earth. My E looked like a C right over. This refers to the Earth. And then you see
this metry part. And you see metry in
things like trigonometry as well and metry or
the metric system. And this comes from
measurement or measure. So when someone's talking
about geometry, the word itself comes from
Earth measurement. And that's not so bad
of a name, because it is such a general subject. Geometry really is
the study and trying to understand how shapes and
space and things that we see relate to each other. So when you start
learning about geometry, you learn about lines and
triangles and circles. And you learn about angles. And we'll define all of
these things more and more precisely as we go
further and further on. But it also encapsulates
things like patterns and three-dimensional shapes. So it's almost
everything that we see. All of the visually mathematical
things that we understand can in some way be
categorized in geometry. Now, with that out of
the way, let's just start from the basics, a basic
starting point from geometry. And then we can just
grow from there. So if we just start at a dot,
that dot right over there, it's just that little point on
that screen right over there. We literally call that a point. And I'll call that a definition. And the fun thing
about mathematics is that you can
make definitions. We could have called
this an armadillo. But we decided to call this
a point, which, I think, makes sense, because it's
what we would call it in just everyday
language as well. That is a point. Now, what's interesting
about a point is that it is just a position. You can't move on a point. If you were at this point and
if you moved in any direction at all, you would no
longer be at that point. So you cannot move on a point. Now, there are differences
between points. For example, that's
one point there. Maybe I have another
point over here. And then I have
another point over here and then another
point over there. And you want to be able to
refer to the different points. And not everyone has the
luxury of a nice colored pen like I do. Otherwise, they could
refer to the green point or the blue point
or the pink point. And so in geometry
to refer to points, we tend to give them labels. And the labels tend
to have letters. So for example, this
could be point A. This could be point B.
This would be point C. And this right over
here could be point D. So someone says,
hey, circle point C. You know which one to circle. You know that you would
have to circle that point right over there. Well, that so far is
kind of interesting. You have these
things called points. You really can't move
around on a point. All they do is
specify a position. What if we want to move
around a little bit more? What if we want to get
from one point to another? So what if we
started at one point and we wanted all of the points,
including that point, that connect that point at
another point, so all of these points right over here. So what would we call this
thing, all of the points that connect A and
B along a straight-- and I'll use everyday
language here-- along a straight line like this? Well, we'll call
this a line segment. In everyday language,
you might call it a line. But we'll call it
a line segment, because we'll see when we
talk in mathematical terms a line means something
slightly different. So this is a line segment. And if we were to
connect D and C, this would also be
another line segment. A line segment. And once again,
because we always don't have the luxury
of colors, this one is clearly the
orange line segment. This is clearly the
yellow line segment. We want to have the labels
for these line segments. And the best way to
label the line segments is with its end points. And that's another word here. So a point is just
literally A or B. But A and B are also the end
points of these line segments, because it starts and ends at
A and B. So let me write this. A and B are end points, another
definition right over here. Once again, we could
have called them aardvarks or end armadillos. But we, as
mathematicians, decided to call them end
points, because that seems to be a good name for it. And once again, we need a way to
label these line segments that have the end points. And what's a better way
to label a line segment than with its actual end points? So we would refer to this
line segment over here, we would put its
end points there. And to show that
it's a line segment, we would draw a line
over it just like that. This line segment down here,
we would write it like this. And we could have just as
easily written it like this. CD with a line over it would've
referred to that same line segment. BA with a line over it would
refer to that same line segment. Now, you might be saying,
well, I'm not satisfied just traveling in between A and B. And this is actually
another interesting idea. When you were just on A,
when you were just on a point and you couldn't
travel at all, you couldn't travel in any
direction without staying on that point, that means you
have zero options to travel in. You can't go up or down, left
or right, in or out of the page and still be on that point. And so that's why we say a
point has zero dimensions. Zero dimensions. Now, all of a sudden,
we have this thing, this line segment here. And this line segment
we can at least go to the left and the right
all along this line segment. We can go towards
A or towards B. So we can go back or
forward in one dimension. So the line segment is a
one-dimensional idea almost or a one-dimensional object. Although these are
more abstract ideas. There is no such thing as
a perfect line segment, because you can't move up
or down on this line segment while being on it, while in
reality, anything that we think is a line segment, even a
stick of some type, a very straight stick or a
string that is taut, that still will have some width. But the geometrical pure
line segment has no width. It only has a length here. So you can only
move along the line. And that's what we say
it's one-dimensional. A point you can't move at all. A line segment you can only
move in that back and forth along that same direction. Now, I just hinted that it
can actually have a length. How do you refer to that? Well, you refer to that by
not writing that line on it. So if I write AB with a
line on top of it like, that means I'm referring
to the actual line segment. Let me do this in a new color. If I say that AB is equal to 5
units-- it might be centimeters or meters or whatever, just
the abstract units 5-- that means that the distance
between A and B is 5, that the length of line
segment AB is, actually, 5. Now, let's keep on extending it. Let's say we want to just
keep going in one direction. So let's say that I start at A. Let me do this in a new color. Let's say I start at A.
And I want to go to D. But I want the option
to keep on going. So I can't go further
in A's direction than A. But I can go further
in D's direction. So this little idea that I
just showed, essentially, it's like a line
segment, but I can keep on going past this
endpoint, we call this a ray. And the starting point for a
ray is called the vertex, not a term that you'll
see too often. You'll see vertex later
on in other contexts. But it's good to know. This is the vertex of the ray. It's not the vertex
of this line segment. So maybe I shouldn't
label it just like that. And what's interesting
about a ray is it's once again a
one-dimensional figure. But you can keep on going
past one of the endpoints. And the way that we
would specify a ray is we would call it AD. And we would put this little
arrow over on top of it to show that it is a ray. And in this case,
it matters the order that we put the letters in. If I put DA as a ray, this
would mean a different ray. That would mean that
we're starting at D. And then we're going past
A. So this is not ray DA. This is ray AD. Now, the last idea that I'm
sure you're thinking about is, well, what if I could keep
on going in both directions? So let's say I could keep going. My diagram is getting messy. So let me introduce
some more points. So let's say I have
point E. And then I have point F right over here. And let's say that I
have this object that goes through both E
and F but just keeps on going in both directions. When we talk in geometry terms,
this is what we call a line. Now, notice a line never ends. You can keep going
in either direction. A line segment does end. It has end points. A line does not. And actually, a line
segment can sometimes be called just a segment. And so you would specify line
EF with these arrows just like that. Now, the thing that you're
going to see most typically when we're studying geometry
are these right over here, because we're going to
be concerned with sides of shapes, distances
between points. And when you're talking about
any of those things, things that have finite
length, things that have an actual length, things
that don't go off forever in one or two
directions, then you are talking about a
segment or a line segment. Now, just to keep
talking about new words that you might confront
in geometry, if we go back talking about a line,
I was drawing a ray. So let's say I have
point X and point Y. And so this is line segment XY. So I could denote
it just like that. If I have another
point, let's say I have another point
right over here. Let's call that point Z. And
I'll introduce another word. X, Y, and Z all lie
on the same line, if you would imagine
that a line could keep going on and
on forever and ever. So we can say that X,
Y, and Z are collinear. So those three
points are collinear. They all sit on the same line. And they also all sit
on line segment XY. Now, let's say we're told
that XZ is equal to ZY and they are all collinear. So that means this is telling
us that the distance between X and Z is the same as the
distance between Z and Y. So sometimes we can
mark it like that. This distance is the same
as that distance over there. So that tells us that
Z is exactly halfway between X and Y. So
in this situation, we would call Z the
midpoint of line segment XY, because it's exactly
halfway between. Now, to finish up, we've
talked about things that have zero
dimensions-- points. We've talked about things that
have one dimension-- a line, a line segment, or a ray. You might say, well,
what has two dimensions? Well, in order to
have two dimensions, that means I can go
backwards and forwards in two different directions. So this page right
here or this video or this screen that
you're looking at is a two-dimensional object. I can go right-left. That is one dimension. Or I could go up-down. And so this surface of the
monitor you're looking at is actually two dimensions. You can go backwards or
forwards in two directions. And things that
are two dimensions, we call them planar. Or we call them planes. So if you took a piece of
paper that extended forever, it just extended in
every direction forever, that in the geometrical
sense was a plane. The piece of paper
itself, the thing that's finite-- and
you'll never see this talked about in a
typical geometry class. But I guess if we were
to draw the analogy, you could call a piece of
paper maybe a plane segment, because it's a segment
of an entire plane. If you had a third
dimension, then you're talking about our
three-dimensional space. In three-dimensional
space, not only could you move left to right along
the screen or up and down, you could also move in
and out of the screen. You also have this dimension
that I'll try to draw. You could go into the screen. Or you could go out of
the screen like that. And as we go into higher
and higher mathematics, although it becomes
very hard to visualize, you'll see that
we can even start to study things that have
more than three dimensions.