Common Core Geometry.Unit #1.Lesson #1.Points, Distances, and Segments

Hello and welcome to common core geometry by E math instruction. My name is Kirk weiler, and today we're going to be doing unit one lesson one on points distances and segments. Now, because it's unit one lesson one, of course, we're beginning a brand new course. So I want to welcome those of you who've watched our algebra videos before to geometry. And before we get into the topics of points distances and segments, I want to talk a little bit about why we study geometry. What is geometry? Why do we study it? Algebra has tons of different applications and you can see them in almost everyday life, whether or not you're doing a line of best fit to try to predict something or setting up an equation in order to solve some kind of function output. But geometry is a little bit different. Now obviously it has applications in art and architecture, but much of what we do in geometry is use reasoning and logic to really discover things about geometric principles. And I'd like to start off with kind of an interesting quote. All right. And I'm going to read this quote to you. It's really rather long, but it goes like this. Since the ancient Greeks, geometry has been the paradigm of truth and ordered knowledge of clear thinking and the rigor of absolute precision of thought. But its real power is that it is also about the world around us. It seems that one of the most fundamental of human scientific intuitions is that the physical world is ultimately geometric. And that to study geometry is in some sense to uncover some of the ultimate essence of the physical world around us. Doctor Pierre's bursal hall, Cambridge institute for geometry. Ultimate essence of the world around us. You see, what we're going to be doing throughout this entire course is giving you tools. Tools to explore that physical world that surrounds you. Most of this will be the two dimensional world, the flat world, but we'll do a little bit of three dimensional geometry towards the end of the course. But for right now, lunch jump into the topic that we're going to do today. Specifically, uh oh, a little bit too far. Let's talk about what a point is. All right? A point is a unique location in physical space. It has no length. No area, no volume. It is zero dimensional. A lot of people incorrectly think that a point has one dimension, but it doesn't. It's zero dimensional. No length area or volume. Now, for us, points have to be drawn as very small circles. But in the reality is, we would never be able to see a point ever because it's got no size. We will always designate points as we have here with a capital letter. A, B, whatever. We can of course have other points like B and C and when we take three points or even more, we can connect them then with an infinite number of points and perhaps create something like a triangle. Or other shapes like that. And we're going to be studying how points, lines, segments, and shapes, all kind of interact in this course. So let's jump right in and do some measurement work. All right? So in our first exercise, what we've got is points a, B, and C, this is in our first exercise. This is our second exercise. Here we go. Exercise one says measure the distance between points a and B symbolized by AB. The distance between B and C symbolized by BC and the distance between points a and C likewise symbolized by AC. Round each of your answers to the nearest 8th of an inch reduce when possible. Now, I know that you've done measurement before, I know you've used a ruler before. But just a little reminder on this because we're going to be doing some length measurements here and there. Let's take a look at how this ruler works. All right? What we want to make sure that we do is we want to put our zero mark as much as we can in the center of point a remember point a should just be a physical location in space. And then we're going to move our ruler up and try to read off as best as we can, the distance from point a to point B all right? And what I noticed here on my gigantic ruler is that distance happens to be two and one 8th inch. All right, what I'd like you to do right now is pause the video and try to figure out the length of the distance between points B and C and also the distance between points a and C to the nearest 8th of an inch. Pause the video now. All right, let's take a look at the distance between points B and C all right, I'm going to rotate my ruler down. Again, get my zero point right in the middle of point B come down here. And I see on this ruler, let me bring the screen down a little bit so you can read it a bit better. That the length between point B and C watch at it almost looks like three and a half. But it is in fact three and a quarter inches. Okay, and finally, we can also do the distance between point a and C by bringing my ruler over and I find that that is at a distance of four and one half inches. All right, move my ruler down to here. So if you found those distances correctly, that's great. Now a little bit of note on measurement throughout the course. All right? Due to the fact that worksheets sometimes shrink or expand when your teacher puts them on the photocopier or just due to the fact that you might not quite have the zero point right in the middle of that dot. You may not get exactly the same measurements as I do. So if you had, let's say, two and a quarter up here instead of two and an 8th, that's probably okay. On the other hand, if you had three and a half instead of two and 8, that's probably not okay. All right? So just watch your measurements and realize that you want to get as close as possible, but they may not be exactly what I have. If you need to pause the video now to write down any of this work, please do. And then we'll move on to the next slide. All right, let's move on. Uh oh, again, too many slides. Here we go. So sometimes three points in space have a very special relationship with each other. And we're going to take a look at that in this next exercise. First, let me read it for you. Exercise two. The points a, B, and C are shown below. Find the values of AB, BC, and AC, round to the nearest tenth of a centimeter, IE a millimeter. What is true about the value of AB plus BC? What does this tell you about the three points? Well, let me measure out the length of AB to the nearest millimeter. Then I'll have you do B C and a C and then we'll talk about what their sun represents and what it tells us. All right, let's jump right into it. Now I'm using millimeters this time. Again, I'm going to put my zero mark right in the middle of point a, rotate down and what I find out is that the length of AB symbolized by simply an a sitting by a B is going to be 66 millimeters, or 6.6 centimeters. Either way. All right? Now what I'd like you to do is measure the length of BC and the length of AC, and then we'll take a look at the results. All right, let's go ahead and do it. Let me take my ruler and kind of move it down to here. Figure out the length of BC. It looks like BC. Is equal to 35 millimeters. And let's take a look at AC. The length of AC. Is equal to a 101 millimeters. Now, what do those three measurements tell us about points a, B, and C or specifically, what does the sum of AB plus BC tell us? Well, if we do my 65 66 millimeters, plus my 35 millimeters, what I find is that it's a 101 millimeters. Exactly the same length as AC. And what that now tells you is that the three points lie in a straight line. They are what are known as collinear. Let's take a look at this a little bit a little bit more generally. Down here. You see, three points in space. Three points that are unique in space can really have one of two relationships with one another. They can either form a triangle like these do, in which case we have a length of 8, a length of 6.7, and a length of 13.6. If I do 8 plus 6.7, I get 14.7. Not the same as this 13.6, right? And that's because those three points form a triangle. On the other hand, if I move point C up so that a, B and C are what is known as co linear. Then I find that the length of AB 8 plus the length of BC 5.7 ends up being the length of AC 13.7. All right? When that happens, what we know is that the points are what is known as co linear. Which I oftentimes misspell and only use one L but I'm going to try my best to spot it correctly here. And it's very, very important. Because we oftentimes think that if we have three points, they're going to lie in a triangle. But we can easily have three points of course that lie in a straight line with one another. And collinear points are going to be very important as we move on. Speaking about moving on, pause the video now if you need to to think about any of the work that we've seen, let me move this back up so that you can see the measurements from before. All right. And then we'll move on to the next problem. All right, let's do it. Ah, my slight advance worked this time. All right. Now, what we've been doing all along, right? Is measuring the distance between two distinct points. We can think of this, though, also is simply the length of a line segment that joins the two points. Just a little bit. Symbolism is going to be important in this course. When we have line segments, IE a starting point and an ending point, and then points filled in between. We're going to always designate those by just putting single little lines above the AB, the CD, et cetera. And again, when we measure the length of a segment, what we're really doing is measuring the distance, the straight line distance between those two points. One of the most fundamental assumptions known as an axiom in geometry is that the shortest distance between two points, which will always refer to as just the distance. But the shortest distance between two points is the length of the line segment straight line segment that connects them. So in this particular problem, exercise three, I ask you to find the lengths of segment AB, CD, and EF shown below. Round your answers to the nearest 8th of an inch and reduce your answers, if possible. In other words, no two fourths of an inch we should have one half of an inch. Why don't you pause the video now and go ahead and do those measurements and then we'll come back and take a look at them. All right, let's fill in the blanks. We're going to the nearest 8th of an inch. Let's take a look at AB. But my ruler right there, move it down, and it looks like AB. And remember, if we're talking about the distance between a and B, we don't have a little, we don't have a little line segment above it. It's just AB without the segment. That's the distance, but that's going to be easy enough. That's one and a quarter inches. On the other hand, the length of CD, if I put my zero mark there and bring it up, it looks like the length of CD is going to be this one's a little bit trickier, but two and 5 eighths of an inch. And finally, if we take a look at the length of EF. Might even be a little bit easier for you than me. Now here's one of course where it doesn't seem to be a perfect length, right? But still, to the nearest 8th of an inch, it certainly seems like it's right around. Three inches. It's not quite three and an 8th. If anything, it's closer to three and one 16th. But since we're rounding to the nearest 8th of an inch, we're going to say three inches on the dot. All right. You probably don't need to pause the video right now to write these down. But if you need to take a moment. All right, let's move on to the next slide. Maybe. Maybe not. There we go. Okay, exercise number four. In the diagram below, points a, B, C, and D are co linear. There is that term again, meaning that they all fall in a straight line. Find the lengths of segments AC and BD to the nearest millimeter. What statement can you make about these two line segments? All right, go ahead and take a moment to do that. Okay, let's go through the measurements. Here we go. We need the length of AC. We'll bring this up, maybe rotate my ruler just a little bit. And it looks like the length of AC. Is 77 millimeters. All right. We also want to know the length of BD. So now we take a look. And BD is also 77 millimeters. Okay. So what statement can we make about these two line segments? Well, the most obvious one is that they're the same length, right? And line segments truly right now have only one measurement that counts. And that's their length. Now, I wouldn't want to say that line segment AC is equal to line segment BD. I don't want to do this, okay? And the reason I don't want to do that is I don't want to say that an elephant's equal to an elephant, a house is equal to a house, right? Because they're not numbers. We only want to use the equality sign when we're talking about numbers. When we talk about geometric objects that share a measurement, what we use is not the equality symbol, raced a little bit more than I wanted. But we use what's called the congruent symbol, which is an equality symbol with a little similarity symbol above it. And we'll talk about that more as we move on. But basically at the end of the day, we can say they have the same length. So that's what you put down. Excellent. They have the same length. Now, exercise number 5 asks us do segments AB and CD have the same measure. AB and CD have the same measure. All right? And then why does this answer make sense given your answer to exercise four? The first thing we should do is answer the first exercise. Do they have the same length? Take a brief moment to do that right now. Okay, let's take a look. Try to bring this down as far as I can. That's about it. Let's measure AB. AB now has a length of 26 millimeters. All right. CD. Has a length. Of 26 millimeters. So I guess the answer to our first question do segments a, B, and CD, have the same measure. The answer is yes. But why does that answer make sense given what we found in the previous problem? Take a minute to think about this. All right, let's talk about it. This is going to be an important thing to understand later on in the course. In the last exercise, we saw that AC, right? And BD, maybe I'll put the little line segment marks above them, they were the same length. And the key is let me go in a different color now just to really highlight this. The key is both of those two segments AC and BD have line segment BC as part of them. Okay? So if we took segment AC and we removed BC, we literally subtracted that segment out, what we'd be left with is segment AB. Likewise, if we took segment BD and we removed BC, we would be left with segment CD. And therefore it makes sense that those two would be the same length because these two. Are the same length. All right? Eventually, we'll talk about what's called partitioning and how to bring that into geometry and logic. But for right now, just understand the idea. Okay. Let's wrap it up. Today, we talked a little bit about the purpose of studying geometry. Why we study it, we talked about the fundamental idea of a point being a physical location in space. We reminded ourselves how to measure distances and lengths using a simple ruler. And then we looked at what it meant for points to be collinear IE all three points or more falling in a straight line. We never really talked about two points being collinear because any two points that are distinct will fall in a line. Okay. Well, we're going to be building on this obviously in our next set of lessons. For now, I just want to thank you for joining me for another common core geometry lesson by E map instruction. My name is Kirk weiler. And as always, keep thinking. And keep solving problems.