Geometry Points, Lines and Planes

Welcome back to Educator.com.0000

This lesson is on points, lines, and planes; we are going to go over each of those.0002

First, let's start with points: all geometric figures consist of points.0010

That means that, whether we have a triangle, a square, a rectangle...we have a line...0017

no matter what we have, it is always going to consist of an infinite number of points.0023

A point is usually named by a capital letter, like this: this is point A.0030

(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.0037

Next, lines: a line passes through two points; so whenever you have two points, you can always draw a line through them.0051

So, a line has at least two points; lines consist of an infinite number of points.0061

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.0070

So, every point on this line is one of the infinite numbers; so we have many, many, many points on this line, not just 2.0079

A line is often named by two points on the line, or by a lowercase script letter.0091

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).0096

Now, this line has arrows at each end; that means it is going continuously forever, infinitely continuous.0112

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.0126

And this is actually supposed to go like this, AB; or it could be BA, because it is going both ways.0140

Line AB...now, when we say line AB, then we don't draw a line above it, like this, because,0153

when we say "line," that takes care of it; we don't have to draw the line, because we are saying "line."0160

Line AB or line BA...this can also be line n in script, or AB with a line above it--a symbol.0167

Next, for planes: a plane is a flat surface that extends indefinitely in all directions.0180

Planes are modeled by four-sided figures; even though this plane is drawn like this, a four-sided figure,0187

it is actually going to go on forever in any direction.0196

If I draw a point here, then I can include that in the plane, because the plane is two-dimensional;0202

so the points could be either on the plane...or it might not be.0210

But they are modeled by four-sided figures; and make sure that it is flat.0218

A plane can be named by a capital script letter or by three non-collinear points in the plane.0225

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.0230

Here are three non-collinear points (non-collinear, meaning that they do not form a straight line):0245

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.0254

Now, it doesn't have to be ABC; it could be plane BCA; it could be plane CBA, CAB...either one is fine.0266

Now, drawing and labeling points, lines, and planes: the first one here: we have a line.0281

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.0290

It is just the first part; this line is line n; I don't have two points on this line labeled,0297

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.0308

So, S, a capital letter--that is how points are labeled: S, or point S, is on n, or line n.0318

I could say that line n contains point S, or I can say that line n passes through S, or point S.0329

Even if it doesn't say plane N or point S, just by the way that the letter is written,0342

how you see the letter, you can determine if it is a point, a line, or a plane.0348

The next one: l and p intersect in R; how do we know what these are?0355

It is lowercase and script; that means that they are names of lines, so line l and line p intersect in R.0362

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.0372

l and p both contain R, meaning that point R is part of line l, and R is part of line p.0381

R is the intersection of l and p; line l and line p--R is the intersection of the two lines.0391

The next one: l (here is l, line l) and T...now, this might be a little hard to see,0402

but when this line goes through the plane, this is where it is touching; so think of poking your pencil through your paper.0412

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.0429

I know it is kind of hard to see, but just think of it that way.0440

So, line l and T, that point, are in plane P--a capital script letter; that is the plane.0443

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.0453

Line m intersects P, the plane, at T; this line right here intersects the plane at that point--that is their intersection point.0465

T is the intersection of m with P; T is a point; point T is the intersection of line m with plane P.0482

Your pencil through your paper--the intersection of a plane with a line--will be a point, and that is point T.0495

The next one: this is a little bit harder to see; I know that it is kind of squished in there.0504

But here we have two planes: this is plane N, and this is plane R.0511

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,0519

and it is also passing through plane N; and on that line are points A and B.0533

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.0544

So, AB is in plane N, and it is in plane R; this line is in both.0560

N and R, both planes, contain line AB; what does that mean?0575

If this line is part of both planes, that means that the line is the intersection of the two planes.0587

Think of when you have two planes intersecting; they are always going to intersect at a line.0592

It is not going to be a single point, like a line and a plane; two planes intersect at a line.0599

We will actually go over that later; N and R intersect in line AB.0605

The line AB is the intersection of N and R; there are different ways to say it.0614

Let's go over some examples: State whether each is best modeled by a point, a line, or a plane.0621

A knot in a rope: the knot...if I have a rope, and I have a knot, well, this knot is like a point.0627

This one is going to be a point.0637

The second one: a piece of cloth: cloth--a four-sided figure--that would be a plane.0642

Number 3: the corner of a desk: if I have a desk, the corner is going to be a point.0653

And a taut piece of thread; this thread is going to be a line.0664

The next example: List all of the possible names for each figure.0677

Line AB: this line can be line n; that is one name.0680

It can be like that, line AB or line BA; it can also be BA in symbols, like that, or BA this way.0694

The next one: Plane N: this is one way to name it.0716

I can also say plane ABC, plane ACB, plane BAC, plane BCA, CAB, and CBA.0721

There are all of the ways that I can label this plane.0746

Refer to the figure to name each.0757

A line passing through point A: there is point A; a line that is passing through is line l.0760

Two points collinear with point D (collinear, meaning that they line up--they form a line):0776

two points collinear with point D, so two points that are on the same line: points B and E.0785

A plane containing lines l and n: well, there isn't a plane that contains lines l and n,0800

because this line l is not part of plane R.0819

How do we know? because it is passing through, so it is like the pencil that you poke through your paper.0824

It is not on the plane; it is just passing through the plane.0832

So, a plane containing lines l and n is not here.0836

If I asked for two lines that plane R contains, I could say plane R contains lines n and...0842

the other one right here; there is no name for it, so I can say line FC.0866

I can write it like that, or I can say and line FC, like that.0876

OK, the next example: Draw and label a figure for each relationship.0881

The first one is point P on line AB.0890

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.0895

CD, the next one: line CD lies in plane R and contains point F.0910

So, I have a plane; line CD lies in plane R (this is plane R) and contains point F; the line contains point F.0918

Points A, B, and C are collinear, but points B, C, and D are non-collinear.0947

OK, that means I can just draw a line first, or I can just draw the points first, points A, B, and C.0953

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.0968

OK, so A, B, and C are collinear, but B, C, and D are non-collinear.0979

OK, the next one: planes D and E intersect in n.0985

Now, this is a line, because it is a lowercase script letter.0990

So, here is one plane; here is another plane; let's label this plane D; this could be plane E.0994

And then, where they intersect, right here--that will be line n.1019

OK, that is it for this lesson; thank you for watching Educator.com.1029