Points Lines & Planes in Geometry

Bam, Mr. Tarrou. In this video, we’re going to start our lessons in Geometry and learn about the three most basic structures in Geometry: the point, the line, and the plane. And run through a lot of definitions like colinear, coplaner, intersections … find out how planes intersect with each other, if they actually even do So, we’re just going to … not do any problems, just lay down a foundation of vacab(ulary) that we’re going to need in our later studies of Geometry. So, the current number of pixels in an iPad, as an example, is over 3.1 million pixels, in an iPad with, what, a 10-inch screen? So, each one of those tiny little pixels … each one of those points … see the tie in there … is so small, or are so small, that they’re nearly impossible to see with the naked eye. Yet, put enough of those together, and you can form lines, planes, and volume if you are talking about 3-dimensional space, which we are not going to be in this video. So, a point has, let’s see here, a point is our simplest figure in Geometry. It has no size … none … no length, no width, no volume. It’s a point defined in space, but, in pure mathematical theory, it sort of doesn’t exist, but yet we’ve defined it. There’s no measurable, like, object there. It’s just a place in space identified. So, it has zero dimension. Now, I’ve identified a couple of points here on the chalk board and we, of course, have to represent if with something that we can see, so we put a dot, which, has size (it’s so wide and so tall), but, you know, I’ve got to be able to see it So, at this point I’m going to identify this as Point A and Point B. Now, with my little dots here, of course, what you might want to also identify is how we are labeling points. We label those with a capitol letter in Geometry. So, we have Point A and Point B here. Now, if we take these points that have no length or width, have no size, but yet, you know, start putting those, start lining them out. Just think of like a pearl necklace or a beaded bracelet, where each of those beads are a point and you start lining up … they’re very very tiny but you start lining them up, and all of a sudden you have this line formed. Or, of course, you know, going back to the iPad, in this 10-inch screen that has over 3 million pixels in this 10-inch screen. Of course, you’re looking at it. You might be watching me on an iPad right now, and there’s a line made up of, you know, thousands, tens of thousands, of these tiny little, almost invisible dots. But yet, line up enough of them and they can start forming a line. These lines extend in two directions, without end. OK? So, it goes indefinitely. So this white … of course, it’s not perfectly straight here, either, but forgive me on my artwork. This dividing line that I’ve drawn on my chalkboard to divide the two sides of notes, really isn’t a line because it ends and starts, so it’s really a line segment, which we’re going to get to. But if you’re talking about a line by definition, it extends in two directions indefinitely. And you only need 2 points to define a line because it does not bend, it does not change direction. So, it extends it 2 directions without end, it has a length, but it’s length is infinity … and Beyond! It has an infinite length because it never ends but it does have length, a measurable, well, the measure of infinity, but yet, no width or no height, so lines are 1-dimensional. I’ve identified this line with the 2 points of A and B. If I wanted to change the direction of this line I’d actually have to change the position of one of these points and then redraw the line through those two points And so we can identify this with notation, as here, as Line AB … I put the two capital letters together and have sort of like a double arrow on top of AB going in both directions (indicating a line). Line BA, and it does not matter the order that those variables are placed, those points are placed in the name because it extends in both directions forever. So, the order of those letters don’t make any difference when you identify what line you are referring to in a figure. Or, a lot of times in your geometric figures they’re going to give you a lower-case letter somewhere at the beginning or end, you know, or near one of the arrows of that line and that means that you can use those to indicate the name of that line. So this is also Line l (lower-case “L”) because I have a lower-case “L” up there near one of the arrows. Now, think of like taking these lines that extend for infinity, which is a collection

of an infinite number of points, and you start laying them out side-by-side, kind of like … think of like someone’s house or maybe a wooden deck and you have these big long planks of wood, these lines (sort of), and you start laying them side-by-side. Now a line has no width, but again, if you lay line by line by line by line, and you can see this on a computer screen as well, where you can draw a line and then right next to it you can draw another one right next to it draw another one. And eventually, they’ll mathematically … their width is non-existent, it has a zero width … but those start building up (sort of trying to mix-in a little bit of theory here without doing a real proof) but, you know, you start laying them down, down, down, and all of a sudden you get this plane Planes extend without bound, but yet again have no height. So, where a line is just going in one direction and has a length, a plane has a length, and then you’re laying those lines, if you will, sideways, so you have width as well. So, you have length and width in a plane. Of course they are also going to extend without bound, and so the length and width of a plane … that’s infinity as well for each of those measurements. So, since a plane extends in 2 directions, actually it extends in all directions, but there’s, you know, no height, so just for simplicity of length and width, since it extends in this 2-direction idea … sort of … really you could say that it extends in an infinite number of directions, but all within the same plane. Hmmn. Then, we see it’s a 2-dimensional object. So it extends without bound, has no height. Planes have an infinite length and width While there was no true edge, we have to draw something, so you normally see planes represented in geometric figures or drawings as a parallelogram. You could draw a rectangle as well, but, you know, we draw a rectangle or, usually, a parallelogram and somewhere in there put a capital letter and thus this is Plane M. Of course, we have to, again, draw something, so we generally make them look like parallelograms. But, indeed, they are not because parallelograms do not extend with length and width forever. Let’s get to the next screen. We’re going to define collinear points, coplaner points, intersections, and so-on. So, let’s get to the next page of our notes Let’s get those definitions started. Space! The Final Frontier. And also, in Geometry, is the set of all possible points. Or the set of all points (excuse me) in the whole universe. Space. Colinear points are all in the same line, so collinear points, remember a line is going to be straight, it doesn’t bend, it doesn’t change direction, just a straight line that extends indefinitely in both directions. Now, it only takes 2 points to define a line because it does not change direction; therefore, it is straight. If I have 3 points that are sharing a line, there’s a lot of theorems that we’re going to learn in Geometry, or a handful of them anyway, to show that a collection of points are collinear. I’m going to pull a reference that you know from Algebra which is just using slope. Because, what I’m saying, you know, a line is straight and does not change direction, slope that you learned in Algebra which is Rise/Run (Rise over Run) describes the direction that a line is taking. So, if I’ve got 3 points here (A, B, and C), then the slope from A to B is equal to the slope from A to C which is equal to the slope from B to C. I’ve collected all the different variations that I can find slope using these 3 points and they’re all the same They’re all equal. So, those three points do all fall within the same line. And, of course, I have to give you a non-example, so these points of D, E, and F, you can see a sharp bend that I would have here if I tried to connect Let’s see, if I tried to connect D to E and then to F, I would certainly not get a figure that appeared to be straight, and therefore, it is not a line. And they’re not collinear. Coplaner points, as you might know or guess, are points that all share or all fall within the same plane. So I’ve just simply drawn a Plane M here and identified 3 points that plane sort of going to this left-right direction no first extended definitely where I have the points of D and F inside that plane and then coming through from a different direction maybe let’s see here just to give you more of a three-dimensional example I’ve got this plane here de and it holds the

point de i’m going to have another plane coming in and intersecting it that has this point at try to identify those with different colors to you know make my diagram most legible hopefully those colors are showing through upon the camera Saudi are in this green plane . f is in the yellow one and therefore these three points together or not coplanar we interrupt this regularly scheduled program to bring a correction because well while i’m correct ninety percent of the time i am wrong the three-percent who cares about that probably use since you’re watching this video to actually learn something so i’m about to say or just said that three points are coplanar in these three points are not that is a problem because three points define applying just think of my cameras on a tripod right now in those three points where the tripod is touching the floor you know that basically those three points are defining the plane of the floor so saying these three points are coplanar is no big deal because well three points to find a plane so my problem with my example i just did is I need a fourth and now let’s make a big deal Hey look at drawing these four points and they’re all the same plane so these are coplanar de nf1 my drawing is then made purposely to attempt to make them appear to be in a separate in separate planes i could draw a third plane that is defined by these three points we kind of come in here at an angle and skips a really quick the kind of ugly on this diagram so let me introduce a fourth . and then I can actually say hey look these four points are not coplanar so let’s uh this one plane here that I said was going in the same direction the board let’s call this d e and let’s put a . g over here and so the points of d e and f are defining this plane that I’ve drawn sort of horizontally and kind of just sort of like a rectangle and then this . f if you see this intersection line it’s kinda like this i’m trying to make this point here in this plane coming out of the board kind of appear to be in front it’s almost like . FB kind of like out here and points d e and g are actually on the plane of the chalkboard okay so now i have four points that are not coplanar I’ve got three points that are on the same player in the same plane and three points to find a place that’s always the case we have d e and g those are in this plane right here and . f is in this diagram not in the same plane it’s on this other plane coming in front and it’s actually sort of out here in space in front of the chalkboard now getting back to the rest video over here we have an intersection just like intersection roads where two roads me on the intersection of blank and blank same thing geometry where two figures cross each other intersect well they intercept so intersection of two figures is a set of all points that are in both figures so we have a line l and a point a that is on that line so it’s an intersection here right here this green . so . a is either in or on the language is interchangeable line L capital letter for the point and the lowercase letter indication for the line don’t forget that or line L passes through a now if I were to introduce a new . like say here is point B not control that . up there near line L but there is no intersection because of course lynelle is not going to that point the way it is drawn if two points or excuse me if two lines intersect the intersect at a point now two lines don’t have to intercept will be a definition coming up later that you’ll actually should recognize from some of your algebra studies parallel parallel lines do not cross they go in the same direction actually you don’t have to have lines be parallel for them not to cross they could be in other planes as well so my forearms are lines they could be got one going in this direction and one going in this direction but this arm is in front of this arm so these lines are not intersecting they could be parallel being the same plane but also not intersect again those that these definitions in mining detailed very specific details are coming up later but if they do intersect they are going to intersect in a point i have plain h2o on and have a couple lines dinner some sort of working with this plane eh I’ve got a line going through it so I’m flying el going through playing aah i’m trying to identify its going through because you see here the line is solid sort of like let’s see well let me get a pencil I can have this you know plane which is my piece of paper and this line for this

line coming through it well if the line is sort of between your view and the plane is going to be solid but when that line comes through the plane and in falls below you’re not really going to see that line but we draw these dotted lines to represent that you know we understand this this blue line is solid it’s passed through the plane and I’m just making a dotted so to show representation minutes you know still there is just sort of behind the plane so that we can’t see it and then of course we make it solid after it the view of it goes beyond the edge of the plane that we draw now again remember planes don’t have an edge but we have to be able to all something so as that line comes back from behind the yellow plane we’re going to make it solid again to indicate that we can clearly see it and then we have this Green Line k which we’re going to draw as being in plain H so we have K like a we know it’s like a because it’s lower case and capital piece that’s going to be a point P we see it right here that’s the intersection point between the line and the plane h-here lowercase K like a and P are art in playing h and line L intersects h at Pete this line k right now is in the plane so again let’s see your paper and my pencil so this could be playing aah my line is literally like in it now you can have a line be above plane as well and not be contained within that plane or intersected that’s not what I have drawn here because we’re defining intersection you know identifying ways the different geometric figures these lines planes and not . can intersect and contain each other up here got two plays drawn I’m trying to of course give some depth to my two-dimensional drawing he’s on this flat chocolate have plain be and I have playing a sort of coming in here at an angle and they are intersecting at a line planes again do not have to intercept I can have a two-dimensional plane here in a two-dimensional plane here they can be parallel and never intersect however these planes are intersecting they’re intersecting in line c of two planes intersect they’re always going to form a line now with the CD with this line segment here it does the planes extend in all directions or well not all directions they don’t there’s no height but the planes do a nurse extended length and width forever you could argue that that’s every direction if you stand there just going to you know so we rotate lookout and the lines extending both directions forever as well but again we have to be able to draw playing somehow so please again for one more time you know like this line this is like a line segment something we defined in a minute but because it’s an interception of two planes it is going to extend in both directions forever so planes a and B intersect in line CD keep that double arrow indicator above the CD if you don’t indicate that it means something else would be the length of a line segment we’re not talking about measuring length and both playing a and playing the contain line CD of these definitions are making sense for you let’s get to the next page which is going to be looks like just an example for our series of questions here that we’re going to use our examples before finishing up the video John come on attended all three dimensional sort of house figure here and what i did was i drew a five-sided figures a pentagon and flat horizontal of Base to vertical sides and a peak in the middle what I attempted to draw this as best I can is I went off of each one of these corners I this one first and with a certain direction for a certain length then what I try to do was that went to the rest of the four corners five total and I try the best i could I can see here there’s a little bit of separation here with these two lines but I tried to go off and make for more lines and all went in exactly the same direction parallel and I also attempted to make them the same length then use a ruler probably should’ve and if you do that well enough if you draw your front face and least that’s what i do this figure draw my front face and then go off each one of the corners in and making parallel equal

line segments and then sort of just connecting the dots in the back remember to put some dotted lines to represent those lines that the back of the figure that if it were solid you would really see figure so that’s the tip on how to try and draw we want two planes that intersect in line are s now RS is this line here that is at the top of our sort of house if you will and all these are an intersection of a lot planes so what two planes that intersect for you know that extended all directions or you know sort of this two-dimensional plane forever what lon date you know that in a form they intersect to form that line RS so you wouldn’t be abie that’s the base of this house this letter right hand side here f b cg is extending vertically forever so that’s not going to intersect line RS it’s the two you know basically two sides of the roof playing our f GS you can you can demo playing with four letters when they’re drawing this fashion don’t have any going to be too confusing to throw more letters in there to be able to identify the plane as a single letter or with the same letter so playing RF GS is coming and leaning in from the right and playing e.r s and see that H back there that plane is leaning over to the right and together they’re coming too forum that line RS so again it’s playing and you can do that with their little pal parallelogram figure as well so playing wasn’t our of GS and plane was going to write P playing ER Sh okay three lines that intersect at hwood see worse h/h is this back corner here so i needed to identify line with two capital letters don’t forget the double layer on top that come together in touch will help to create or intersect at that point H so RF is in the front of this house that’s not going to intersect HS in the back and so we have eh2 wine eh double arrows line s H and last but not least we have line d.h now you see there’s no again just one more time did none of these lines have arrows extending both directions on they look like line segments where they stopped but this is this diagram is supposed to represent we are talking about the intersection of planes so that’s why they’re being said being described as lines and therefore be extending extending on these lines extending both directions forever because the planes extended forever as well three planes that intersect at seeds of three planes can intersect if they do intersect three planes intersect in a line they can also intersect in a plate or a penny . so let’s see here . c is the bottom right-hand corner and that is that . here at sea is made up with the plane on the side of this house if you will the base or bat bottom of the house and then finally this backplane that we have drawn looking at the moment like a pentagram our Pentagon using so we have let’s see here playing i’m going to use it same sort of generic paulo parallelogram notation for all three of these planes that doesn’t matter that the back has five sides so on let’s see here bfg see and plain a b c d and playing the h.s GC please note as well as i’m naming off these planes I’m not trying to jump around i’m not saying that this this plane is a see also here B and then back to deep I’m going around the corners and sequence it’s really very important even more really when you’re identifying like an actual of parallelogram or rectangle will be dealing with those later okay by the way three points to find a plane so one could argue that I could maybe

name the bottom with say it’s you know play a DC but that would be confusing because it would kind of a kind of seemed like that there was only three points are so i’m using all four corners but three points due to find a plane and two points to find line our ad and C coplanar those three points in the same plate let’s see here a is right here we have D which is sort of follow along the this the sort of bottom left-hand line here so we have a and E those are sposa in the same line and see well let’s see here a d.c all of these points are helping to make up the bottom plane of this sort of house like figure so yes indeed those three points are coplanar now what about our s and C ours here in the front of this house if you will s is in the back and see is also within the backplane so the backplane here contains the points of cg s H and D but . r is in the front of this this three-dimensional sort of house figure again it is hard to see on a chalkboard which is two-dimensional that this is what we’re trying again show some depth with the fact that all the angles are kind of like in this sort of parallelogram Leanback sort of kind of diagram and of course with all the Don lines again trying to emphasize the fact this is a three-dimensional object so r s in the arse in the front plane of the house s and C on the back so the answer is no they are not named two planes that do not intercept two planes if you will that are parallel let’s see here the right side of the house intersects with the bottom plane the front plane coming down here Berkeley is going to intersect with this one that’s sort of flat we need to that do not intercept well with this we want to look at either the front and back plane because they’re looking you know sort of like this we’re not going to intersect so the front plane and the backplane are not going to intersect so planes AE RF and be that plane is not going to intersect DHS G&C so that would be to pop that would be a possible answer the two pair together another possibility would be the left side and the right side because those are well less this is you know converge in the back those are not going to intersect as well it does not appear to be in this diagram so playing ke h d ok that’s the left side of this house if you will and the right side dfgh that would be another possible set of answers for the last question your name two planes that don’t intersect so i hope i hope you understand a little bit on points planes and lines that’s in my videos up on the strip go do your homework