WEBVTT mathematics/geometry/pyo
00:00:00.000 --> 00:00:02.300
Welcome back to Educator.com.
00:00:02.300 --> 00:00:10.100
This lesson is on points, lines, and planes; we are going to go over each of those.
00:00:10.100 --> 00:00:16.700
First, let's start with points: all geometric figures consist of points.
00:00:16.700 --> 00:00:23.100
That means that, whether we have a triangle, a square, a rectangle...we have a line...
00:00:23.100 --> 00:00:29.800
no matter what we have, it is always going to consist of an infinite number of points.
00:00:29.800 --> 00:00:37.500
A point is usually named by a capital letter, like this: this is point A.
00:00:37.500 --> 00:00:51.400
(x,y), that point right there, the ordered pair, is labeled A; it is called point A; that is how it is named--point A--by capital letter.
00:00:51.400 --> 00:01:01.300
Next, lines: a line passes through two points; so whenever you have two points, you can always draw a line through them.
00:01:01.300 --> 00:01:10.000
So, a line has at least two points; lines consist of an infinite number of points.
00:01:10.000 --> 00:01:18.800
With this line here, line *n*, I have two points labeled here, A and B; but a line consists of an infinite number of points.
00:01:18.800 --> 00:01:31.100
So, every point on this line is one of the infinite numbers; so we have many, many, many points on this line, not just 2.
00:01:31.100 --> 00:01:36.500
A line is often named by two points on the line, or by a lowercase script letter.
00:01:36.500 --> 00:01:51.700
The way we label it, or the way we name it: this is an n in script; I can call this line *n*, or I can call it line AB (any two points).
00:01:51.700 --> 00:02:06.600
Now, this line has arrows at each end; that means it is going continuously forever, infinitely continuous.
00:02:06.600 --> 00:02:20.000
It never stops; since it is going in both directions, I can say that this is line AB, or line BA; this can also be BA.
00:02:20.000 --> 00:02:33.000
And this is actually supposed to go like this, AB; or it could be BA, because it is going both ways.
00:02:33.000 --> 00:02:40.600
Line AB...now, when we say line AB, then we don't draw a line above it, like this, because,
00:02:40.600 --> 00:02:46.700
when we say "line," that takes care of it; we don't have to draw the line, because we are saying "line."
00:02:46.700 --> 00:03:00.400
Line AB or line BA...this can also be line *n* in script, or AB with a line above it--a symbol.
00:03:00.400 --> 00:03:07.500
Next, for planes: a plane is a flat surface that extends indefinitely in all directions.
00:03:07.500 --> 00:03:16.000
Planes are modeled by four-sided figures; even though this plane is drawn like this, a four-sided figure,
00:03:16.000 --> 00:03:22.200
it is actually going to go on forever in any direction.
00:03:22.200 --> 00:03:30.000
If I draw a point here, then I can include that in the plane, because the plane is two-dimensional;
00:03:30.000 --> 00:03:37.700
so the points could be either on the plane...or it might not be.
00:03:37.700 --> 00:03:45.100
But they are modeled by four-sided figures; and make sure that it is flat.
00:03:45.100 --> 00:03:50.400
A plane can be named by a capital script letter or by three non-collinear points in the plane.
00:03:50.400 --> 00:04:04.700
So, this whole plane is called *N*; we can name this *N*, by a capital script letter, or by three non-collinear points in the plane.
00:04:04.700 --> 00:04:13.700
Here are three non-collinear points (non-collinear, meaning that they do not form a straight line):
00:04:13.700 --> 00:04:26.600
it could be plane *N* (the whole thing is titled *N*, so it could be plane *N*), or it could be plane ABC: plane *N* or plane ABC.
00:04:26.600 --> 00:04:41.500
Now, it doesn't have to be ABC; it could be plane BCA; it could be plane CBA, CAB...either one is fine.
00:04:41.500 --> 00:04:50.100
Now, drawing and labeling points, lines, and planes: the first one here: we have a line.
00:04:50.100 --> 00:04:57.200
Now, I know that this is kind of hard to see, because there is so much on this slide; but just take a look at this right here.
00:04:57.200 --> 00:05:08.200
It is just the first part; this line is line *n*; I don't have two points on this line labeled,
00:05:08.200 --> 00:05:18.300
so I can't name this line by its points; I can't call it line S; it has to be line *n*; that would be the only name for it.
00:05:18.300 --> 00:05:29.600
So, S, a capital letter--that is how points are labeled: S, or point S, is on *n*, or line *n*.
00:05:29.600 --> 00:05:41.900
I could say that line *n* contains point S, or I can say that line *n* passes through S, or point S.
00:05:41.900 --> 00:05:48.400
Even if it doesn't say plane *N* or point S, just by the way that the letter is written,
00:05:48.400 --> 00:05:55.600
how you see the letter, you can determine if it is a point, a line, or a plane.
00:05:55.600 --> 00:06:02.200
The next one: *l* and *p* intersect in R; how do we know what these are?
00:06:02.200 --> 00:06:12.100
It is lowercase and script; that means that they are names of lines, so line *l* and line *p* intersect in R.
00:06:12.100 --> 00:06:21.300
It is just a capital letter, not scripted, so it is just a point, R; so *l* and *p* intersect in R; they intersect at point R.
00:06:21.300 --> 00:06:31.500
*l* and *p* both contain R, meaning that point R is part of line *l*, and R is part of line *p*.
00:06:31.500 --> 00:06:42.500
R is the intersection of *l* and *p*; line *l* and line *p*--R is the intersection of the two lines.
00:06:42.500 --> 00:06:52.600
The next one: *l* (here is *l*, line *l*) and T...now, this might be a little hard to see,
00:06:52.600 --> 00:07:09.600
but when this line goes through the plane, this is where it is touching; so think of poking your pencil through your paper.
00:07:09.600 --> 00:07:19.700
Right where you poke it through, if you leave your pencil through the paper, that point where your pencil is touching the paper--that would be point T.
00:07:19.700 --> 00:07:22.900
I know it is kind of hard to see, but just think of it that way.
00:07:22.900 --> 00:07:32.800
So, line *l* and T, that point, are in plane *P*--a capital script letter; that is the plane.
00:07:32.800 --> 00:07:45.400
*P* contains point T and line *l*; line *l* is just going sideways, so if you just drew a line on the paper, then that would be line *l*.
00:07:45.400 --> 00:08:02.200
Line *m* intersects *P*, the plane, at T; this line right here intersects the plane at that point--that is their intersection point.
00:08:02.200 --> 00:08:15.200
T is the intersection of *m* with *P*; T is a point; point T is the intersection of line *m* with plane *P*.
00:08:15.200 --> 00:08:23.800
Your pencil through your paper--the intersection of a plane with a line--will be a point, and that is point T.
00:08:23.800 --> 00:08:31.000
The next one: this is a little bit harder to see; I know that it is kind of squished in there.
00:08:31.000 --> 00:08:39.100
But here we have two planes: this is plane *N*, and this is plane *R*.
00:08:39.100 --> 00:08:53.500
We have a line that is the intersection of *R* and *N*; so if you look at this line, this line is passing through plane *R*,
00:08:53.500 --> 00:09:03.900
and it is also passing through plane *N*; and on that line are points A and B.
00:09:03.900 --> 00:09:20.600
OK, line AB...the reason why this is labeled line AB is because there is no name for this line; so you just have to name it by any two points on the line.
00:09:20.600 --> 00:09:34.800
So, AB is in plane *N*, and it is in plane *R*; this line is in both.
00:09:34.800 --> 00:09:46.900
*N* and *R*, both planes, contain line AB; what does that mean?
00:09:46.900 --> 00:09:52.500
If this line is part of both planes, that means that the line is the intersection of the two planes.
00:09:52.500 --> 00:09:58.900
Think of when you have two planes intersecting; they are always going to intersect at a line.
00:09:58.900 --> 00:10:05.000
It is not going to be a single point, like a line and a plane; two planes intersect at a line.
00:10:05.000 --> 00:10:13.800
We will actually go over that later; *N* and *R* intersect in line AB.
00:10:13.800 --> 00:10:21.000
The line AB is the intersection of *N* and *R*; there are different ways to say it.
00:10:21.000 --> 00:10:27.600
Let's go over some examples: State whether each is best modeled by a point, a line, or a plane.
00:10:27.600 --> 00:10:37.100
A knot in a rope: the knot...if I have a rope, and I have a knot, well, this knot is like a point.
00:10:37.100 --> 00:10:41.800
This one is going to be a point.
00:10:41.800 --> 00:10:53.000
The second one: a piece of cloth: cloth--a four-sided figure--that would be a plane.
00:10:53.000 --> 00:11:04.000
Number 3: the corner of a desk: if I have a desk, the corner is going to be a point.
00:11:04.000 --> 00:11:16.900
And a taut piece of thread; this thread is going to be a line.
00:11:16.900 --> 00:11:19.900
The next example: List all of the possible names for each figure.
00:11:19.900 --> 00:11:34.500
Line AB: this line can be line *n*; that is one name.
00:11:34.500 --> 00:11:55.900
It can be like that, line AB or line BA; it can also be BA in symbols, like that, or BA this way.
00:11:55.900 --> 00:12:01.100
The next one: Plane *N*: this is one way to name it.
00:12:01.100 --> 00:12:26.600
I can also say plane ABC, plane ACB, plane BAC, plane BCA, CAB, and CBA.
00:12:26.600 --> 00:12:37.200
There are all of the ways that I can label this plane.
00:12:37.200 --> 00:12:40.400
Refer to the figure to name each.
00:12:40.400 --> 00:12:56.000
A line passing through point A: there is point A; a line that is passing through is line *l*.
00:12:56.000 --> 00:13:04.800
Two points collinear with point D (collinear, meaning that they line up--they form a line):
00:13:04.800 --> 00:13:20.500
two points collinear with point D, so two points that are on the same line: points B and E.
00:13:20.500 --> 00:13:39.100
A plane containing lines *l* and *n*: well, there isn't a plane that contains lines *l* and *n*,
00:13:39.100 --> 00:13:43.900
because this line l is not part of plane *R*.
00:13:43.900 --> 00:13:52.300
How do we know? because it is passing through, so it is like the pencil that you poke through your paper.
00:13:52.300 --> 00:13:56.000
It is not on the plane; it is just passing through the plane.
00:13:56.000 --> 00:14:01.900
So, a plane containing lines *l* and *n* is not here.
00:14:01.900 --> 00:14:26.200
If I asked for two lines that plane *R* contains, I could say plane *R* contains lines *n* and...
00:14:26.200 --> 00:14:35.700
the other one right here; there is no name for it, so I can say line FC.
00:14:35.700 --> 00:14:41.200
I can write it like that, or I can say and line FC, like that.
00:14:41.200 --> 00:14:50.400
OK, the next example: Draw and label a figure for each relationship.
00:14:50.400 --> 00:14:54.700
The first one is point P on line AB.
00:14:54.700 --> 00:15:10.100
It is a line...draw a line AB; there is AB, and point P is on the line, so we can draw it like that.
00:15:10.100 --> 00:15:17.700
CD, the next one: line CD lies in plane *R* and contains point F.
00:15:17.700 --> 00:15:46.700
So, I have a plane; line CD lies in plane *R* (this is plane *R*) and contains point F; the line contains point F.
00:15:46.700 --> 00:15:53.500
Points A, B, and C are collinear, but points B, C, and D are non-collinear.
00:15:53.500 --> 00:16:08.600
OK, that means I can just draw a line first, or I can just draw the points first, points A, B, and C.
00:16:08.600 --> 00:16:18.900
They are collinear, but points B, C (there are B and C), and D are non-collinear; so I can just draw D somewhere not on the line.
00:16:18.900 --> 00:16:24.800
OK, so A, B, and C are collinear, but B, C, and D are non-collinear.
00:16:24.800 --> 00:16:30.200
OK, the next one: planes *D* and *E* intersect in *n*.
00:16:30.200 --> 00:16:34.600
Now, this is a line, because it is a lowercase script letter.
00:16:34.600 --> 00:16:58.800
So, here is one plane; here is another plane; let's label this plane *D*; this could be plane *E*.
00:16:58.800 --> 00:17:09.500
And then, where they intersect, right here--that will be line *n*.
00:17:09.500 --> 00:17:17.000
OK, that is it for this lesson; thank you for watching Educator.com.