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.