Common Core Geometry Unit 7 Lesson 3 dilations and Angles

Hello and welcome to another common core geometry lesson by E math instruction. My name is Kirk weiler and today we'll be doing unit number 7 lesson number three on dilations and angles. So in the last lesson, we looked at dilations actually the last two lessons we've looked at dilations, both in and out of the coordinate plane. And we've seen a lot of important properties that they have. Today, we're going to see the third and perhaps most important property of dilations. But before we do that, let's review what we've seen already with them. All right. So we're not going to review the technical definition of a dilation. Maybe a little bit. But remember, with a dilation, you've always got some center point here I'm calling it C and what I'm doing here, and this is what we saw in the last couple of lessons, is I'm taking a line segment a, B, I'm dilating it from point C as the center using some kind of a scaling factor typically denoted by the letter K if K is greater than one, then what happens is like in this picture, a prime moves farther away from C but along the same ray that a was on. And B prime as well, moves farther away from C all right, if K is between zero and one, then it moves closer. It gets compressed in the image. And what we saw was that if we dilate a line segment like AB, what happens is that a prime B prime gets stretched by that same factor K so if K is twice two, then a prime B prime is twice as long as AB, or if K was one half, then a prime B prime would be down here, and it would be half the length of AB. All right? So that's one thing that we saw. And very important, very important for what we're going to do today, the resulting image that image segment is parallel to the original segment, exceptionally important. Now remember, this is all based on the idea that the center point wherever that is, the center point is not on the line segment that we're dilating. Recall that if we do have a line segment that we're dilating, like let's say a B and we've got the center point C, then what happens is it just becomes the same line segment, or it doesn't become the same line segment. But the new line segment just lies on top of the old one. It just kind of stretches it out. That is a gorgeous picture there. So anyway, let's now take a look at the third exceptionally important exceptionally important property of dilations. All right, the third property is so important it gets this huge heading. All right, let's take a look at exercise one. Exercise one in the following diagram, angle BAC, has been dilated with a center point of D and a scale factor of two. All right, so understand here. Right. I got my center at D, I've got this angle. CAB or BAC either one. And again, remember how dilations work, the idea is that I would connect D to a right with some kind of array like that. And I'm using a dilation factor of two, but that's kind of completely irrelevant. And the idea now is that a prime is twice as far from D as a was, I could do the same thing with C, the same thing with B, of course, then giving me B prime or C prime a prime B prime, and I get my new angle. I'm going to just get rid of this really quick. I want that picture that way. Now, letter a very, very important says, list all parallel rays shown in the diagram. Why don't you go ahead and do that? List all parallel rays in this diagram. All right, let's go through it. I'm going to just move my protractor really quick. Well, based on those first two properties of dilations, right? I know that ray AC when it's dilated, its image, a prime C prime, must be parallel to AC, and they look parallel, right? That is parallel to that. So I'm going to say a C is parallel to a prime C prime. All right, so that's one set. But likewise, right? AB, once it's dilated, would have to be parallel to a prime B prime. Then again, it kind of looks that way. And that's it. But what's really cool is we know a lot of different things about parallel lines. Before we get into them and why they're important in this situation, let's take a look at letter B all right, letter B says, using your protractor, what are the measures of angle one and angle three? So what I'd like you to do is I'd like to take you to take your protractor, okay? And I'd like you to measure angle one and measure angle three. And of course, just keep in mind that the measure of angle one in the measure of angle three are the measure of angle BAC and B prime a prime C prime. For right now, don't worry about the measure of angle two. Just measure angles one and three. Let's see what you get. All right. Let's take a look. So kind of come back up here a little bit so we can see this a bit better. I bring my protractor up at the center at the center of the circle. Maybe stretch this out a little bit better so you can see it. And it appears that the measure of angle one is 35°. All right, so that's simple enough. Let's bring it down here now. Again, put the center right there. And again, 35°. I'm going to shrink this thing down a little bit. All right, both of these angles are 35° angles. And this really illustrates the third very important property of dilations, which is that dilations are angle preserving. They're obviously not length preserving. I mean, not unless K is equal to one, but that would be a little bit silly. But they're definitely angle preserving. And we'll prove that in just a minute, all right? But the plain fact is when I dilated angle BAC, which had a measure of 35°, my new angle B prime a prime C prime also had a measure of 35°. Dilations don't change angles. All right? Now let's prove that. And it's kind of cool. The proof is sort of awesome. So let's take a look at this diagram. Letter C says give a reason based on the second property of dilations. The second property of dilations for why these two measures will always be the same. Refer to the diagram above and use proper terminology regarding parallel lines. So just as a side comment, the second top property of dilations is that idea that ray AC is parallel to ray a prime C prime. And as well, AB AB is parallel to ray a prime B prime. So this is that second property. That parallelism that is produced when we dilate a line segment a line or array. So now Y, Y, based on the fact that this is parallel to this, and this is parallel to this. Why does angle one have to be the same as angle three? Pause the video now and think about this a little bit and see if you can come up with a reason. All right, let's talk about it. And this is really kind of cool. But this is where we have to bring angle to it. All right? So based on the fact that a, C is parallel to a prime C prime. Based on the fact that this is parallel to this, angle one and angle two have to be congruent. Angle one must be congruent to angle two. They're corresponding angles. So they're corresponding angles, right? They're in the same place on these sort of parallel lines. But likewise, right? Because now AB is parallel to a prime B prime. Angle two must be congruent to angle three. And again, it's for the same reason. Their corresponding angles. Sorry about that. I'm running out of room. Again, they're in the same spot. The transversal is a prime C prime. Cutting across AB and a prime B prime. And that leaves three and two being corresponding angles, and therefore being congruent. But that means, right? By substitution. Or by the substitution property, angle one must be congruent to angle three. All right? So in fact, the third property of dilations, which says that angles are preserved when we dilate them. All right, based on that third property comes from the second property, which says that when we dilate a line segment or array or align itself, we get parallel lines. Which is kind of cool. All right. Let's illustrate this third property. One more time, here is that third property. Dilations are angle preserving. In other words, the angles of geometric of a geometric object do not change their measure when dilated. And this is one of those things which is absolutely ideal to look at on geo gebra. Here I've got the same kind of scenario. I've got a center point D I've got angle a, B, C, and a prime B prime C prime, kind of random measures, but we can see with a strange dilation factor of 1.8, right? This angle 83.12° just gets reproduced. Again, it doesn't particularly matter where our center is, right? No matter where our center is, that angle remains the same, and it doesn't particularly matter whether our dilation factor is bigger than one or less than one. Let me stretch it back out. There are a little bit. And I'm not trying to fool you, even with the angle itself, what we can see is that no matter what, those two angles remain the same. Again, this is an extremely important property. I would say that this is probably the most important property of dilations. The fact that they preserve the angle measurements of the geometric object that gets dilated. All right? And let's see some consequences of that in the rest of the lesson. Here we go. Now, one way that this idea is tested is in a multiple choice scenario like the following. And let's take a look at it before we move on to more sophisticated problems. Exercise number two if triangle DEF is dilated by a factor of 5, which of the following statements would be true. Measure of angle D prime is 5 times measure of angle D, DE is one 5th of D prime E prime, measure of angle E is one 5th of measure of angle E prime and EF is 5 times E prime F prime. All right, so here's the idea of this particular problem, right? This particular problem is getting you to think about when you dilate something by a factor of 5. What measurements change and what measurements don't change. And the plain fact is, if we've got some triangle, let me call it D, E, F all right, and it doesn't particularly matter where the center is, the center could be anywhere. The plane fact is, when that thing then gets dilated by a factor of 5. Giving us D prime E prime and F prime, angle E is the same as angle E prime, angle D is the same as angle D prime and angle F is the same as angle F prime. The angles of this triangle don't change under the dilation. So these two are out. In fact, the measure of angle D prime is the same as the measure of angle D and the measure of angle E prime is the same as the measure of angle E those two are out. But what a lot of students think dilations are as they multiply everything by 5, you know, or they divide everything by 5. But let's take a look at the last two choices, right? Here, this one says DE, this length is one 5th of this length. All right? And that's actually our winner. Now granted, the way many people would want to think about it is that D prime E prime is 5 times DE. And that's actually the first property of dilations. The idea that when we dilate a line segment, it gets larger by that scale factor no matter where the center of dilation is, we'd have to rearrange this a little bit to get the fact that DE is one 5th D prime E prime, but they're really the same thing. This one is tempting, right? EF is 5 times E prime F prime. But the plane fact is E prime F prime, which is down here as much larger than EF, right? Here it would imply that EF is 5 times the size of E prime F prime. And that's just not the case. So here we've got choice two. Very, very important. Dilations do not change the angles in the object. And what that means is that the object has the same shape, but a different size. Again, we're going to always pretty much assume in a dilation that K isn't one. If K is equal to one, then you've got an object that's actually just identical to the original. It's congruent to it, not whatever else. All right? So let's keep going. Let's take a look at exercise number three. This one's kind of a fun one. All right, in the diagram below, triangle E prime F prime G prime is the image of EFG. After a dilation of an unknown scale factor with an unknown center point. Oh, wow. Okay, so I don't know what the K is. And I don't know what the center point is. I don't know, maybe call it C okay, first things first. It says using tracing paper, verify that the angles of EFG are congruent to the angles of E prime F prime G prime. And this is kind of cool. So what I'd like you to do is to do this along with me. So that's going to take a little bit of pause by you. What I'd like you to do is pause the video right now. Take some tracing paper out and trace out carefully triangle EFG, the one right here that I've got traced out in red. And then I'll show you how to verify that the angle is the same. But take a moment to do that. All right, this is really cool. All right. Remember, angles are just all about sort of rays that extend forever. So what's neat is I can literally take this triangle once I've traced it out, put it right up in there, and notice, again, the idea is that the two rays just have to lie out long each other. So obviously those two angles match up, then if I bring this down here, those two angles match up, and if I bring this over here, these two angles match up. Now, by the way, just one last thing, I'm going to kind of rewind here for a moment. I'm just going to let that sit like that for a moment and kind of go backwards really quick. Bam, bam, bam, bam, bam. All the way back, not all the way back right here. That should look very familiar to that picture. Right? Looks very, very similar to that one. And we'll kind of come back to that in a little bit, but let me get all the way back up to 6. So we verified that all the angles are the same. All we had to do was trace this out and make sure that one angle sits right on top of the other one. Maybe I'll move this over here now. Okay. Now, letter B says or asks us to graphically determine the location of the center of the dilation, mark this point D all right, so maybe I should go back up to here and instead of my center being C, it'll be D leave all construction marks. Now this is something that you could easily be asked to do on a standardized exam. And it really tests to see if you understand sort of how dilations really occur. And remember, the way a dilation would really occur. You know, if I've got some center point, I'll call it D now, because that's what we're calling the center here. And I've got, let's say, point E, right? If I dilate it, then the idea is that I draw out array that connects D to E, and then I stretch out E or shrink it somehow along this ray. All right? And just knowing that is enough for us to be able to figure out where the center of dilation is. All right. And here's why. If I take my ruler, and I connect, let's say F with F prime, the center of dilation has to lie somewhere along here. All right. Somewhere along that ray, or that line, or that segment, whatever you want to think about it as, the center of dilation has to be somewhere on there. So how can I figure out where it lies? Take a moment and see if you can figure that out. All right, did you figure it out? I mean, it's got to lie somewhere along here, but then again, it also has to lie. Somewhere along the line segment that connects E and E prime. It's also got a lie along that segment. And therefore, the center of dilation has to lie right there. All right? Now, just for good measure, if you want, you could also check by taking point G and G prime and connecting them and hope like mad that you did a good job on the last couple. And you see that yeah, it also intersects right there. There's my center of dilation. All right? Easy peasy. You don't need that third line segment, by the way. You could do any two and where they intersect has to be the center of dilation. Now, let's take a look at letter C because we've got the center of dilation, but what we don't have yet is we don't have, let's see if I can get this thing into shape. I don't have the scaling factor. Now, I want you to think about this just for a minute. Think about whether the scaling factor is bigger than one or less than one. Well, hopefully you're saying, wow, the scale factor is bigger than one. And the reason it's bigger than one is that E prime F prime G prime is quite obviously bigger than EFG. So it's all about am I expanding the geometric object in which case K is larger than one, or am I shrinking it? Am I compressing it? Compacting it, then K would be between zero and one. Now letter C says using your ruler to measure. So now it's not just a straight edge. It's a measurement device. Calculate the scale factor to the nearest tenth. Verify it with at least two sets of sides. There may be errors involved to the rounding of the side lengths. All right. Now, remember, we can always figure out the scaling factor by just simply taking a length from the newer picture from the image and dividing it by a length from the smaller picture. All right? So one way to go about finding the scale factor, if I can actually get my ruler, come back here, move for me. There we go. One way to figure out the scale factor is coming in here and saying, all right. I'm going to measure E prime F prime. Okay, and I find that E prime F prime with a little bit of measurement error, maybe thrown in is 28 millimeters. So E prime F prime, let's say is 28 millimeters. And now let me measure EF. All right. Bring this down here. And EF I find to be, let's say it's at 18. Yeah. 18 millimeters. A little bit hard to read this particular ruler, but no, that's 18. One more time on E prime F prime. And yeah, I like that being 28. Now, how do I actually calculate the scale factor from this? Well, K will always be the newer measurement divided by the older measurement. So that's going to be 28 divided by 18, and this could get a little ugly. What do they say? They say, calculate scale factor to the nearest tenth, easy enough, take my calculator, take my calculator, do 28 divided by 18, nope. Let's clear that out. 28 divided by 18 equals, I like it. All right, we'll just round that to 1.6. So what I'd like you to do really quickly is do one more set of measurements, do that division. And see if it comes out to near 1.6. If you get anything from the 1.5 to the 1.7 range, you're probably perfectly fine. But go ahead. And then we'll do one more as well too. All right, let's go through it. Again, you can do a variety variety of different lengths on this. A tons and tons of different lengths. But why not we go ahead and measure F prime G prime? F prime G prime, I'm going to say F prime G prime is 50 5 millimeters. So F prime G prime is 55 millimeters. Now I can measure FG. 36, so let's go. FG is 36, 37, something like that. The meters not meters. All right, and then that means that my K must be new divided by old. 55 divided by 36 is clear this out. 55 divided by 36. Eh, perfect. I know maybe not perfect, but pretty close. Now, it's not a great surprise that I got 1.6 in one case and 1.5 in the other case. That's all due to little rounding errors here and there. If we had the exact measurements, those K's would be precisely the same. And again, we got a K that's larger than one, which means that this geometric object got larger than that one. By the way, did you notice that when I had my ruler up here for F prime G prime? And then I moved it down here for FG. I didn't have to reorient the ruler. I didn't have to tilt it more in either way. And that's because of that parallelism, right? The same thing happened with my E prime F prime in my EF. I didn't have to tilt it at all either after I had gotten the original one because the two line segments are parallel. All right, let's keep going. Last problem. Here we go. Exercise number four. In the diagram of triangle ABC below, D and E have been located on AB and AC, such that DE is parallel to BC. All right, so I know that those two line segments are parallel. Letter a says give a dilation, give a dilation that maps BC onto DE. So I want to give a dilation. Which involves two things. A dilation always involves the center point and a scaling factor. Center point and a scaling factor. All right? And I want to make it such that BC would get mapped to B prime C prime right on top of DE. So think about that. What would we do? All right, let's talk about it. Well, not surprisingly, we could use a center. Of point a all right. We'll use a center of point a and the idea then, of course, is if I want to dilate B if I want to dilate B so that it shrinks down to D so that it pulls back to D and I want to dilate C all right, so that it shrinks back to E, I need a scaling factor. A is good. A is good where it is. But what should my scaling factor be? In this situation because we have no measurements whatsoever. What we have to do is we just have to give a ratio of side lengths, okay? And again, there are a ton of them that would work just fine, all right? But probably the easiest one to do, remember, the K is always new divided by old. Well, the old length, the old length that I had was BC. The new length that I want is DE. All right. So if I now dilate BC by this constant, think about it for a second. If I took BE, and I multiplied it by DE divided by BE, the B's would cancel, and my new length will be DE. Right? Now it's going to work. The whole justify thing that this will work. Because B prime whoops B prime C prime will equal DE. It'll be the same length. And we'll be. Parallel to BC so must lie. On top of DE. Hey, hi, red. No, that wasps. I've seen the red pen come out. This is so cool. So we're guaranteed right. When we dilate using a as a center, and we dilate B, it's going to lie somewhere on here. Likewise with C, if we dilate C with the center of a, it's going to lie somewhere on here. But if we dilate BC, with this scale factor of DE divided by BC, then it's guaranteed that as it comes down, it's going to be the same length as DE. Now, the other thing that's kind of critical, though, is that because DE is parallel to BC, they're going to have to land on top of each other because they're the same length and they're parallel to each other. All right, if for some reason, DE wasn't parallel to BC, then it would actually be impossible to map BC on top of DE using a dilation, using a dilation with a center of a, all right? But let's keep going. Letter B if triangle ABC, the larger one, triangle ABC, was transformed using the dilation specified in a, then explain why a prime B prime C prime would be congruent to a D, E all right? This is kind of cool, right? So now suddenly we've got some congruence in here. Why is it that if we dilate the entire triangle? ABC using a center of a, all right? Using a scale factor of DE divided by BC, why would the resulting triangle a prime B prime C prime be congruent to ADE? Well, let's talk about it. This is kind of cool. They'd be congruent because a would map to a. Wouldn't go anywhere. It's the center of dilation and the center of dilation doesn't move underneath that dilation. So a would start a stay at a, it would map to a B. With map to D. And C weird. C. Would map to E so all three vertices. Of a prime B prime C prime. Let me throw a drawing on there of triangle a prime B prime C prime. Would fall on triangle a. DE. And that's really remember when we were looking at congruence way back in unit two. That's how we established congruence. We established congruence by showing that there was some kind of a mapping that would map all three vertices of one triangle on top of all three vertices of the other triangle. So given that this dilation would leave a right where it is and would map B down onto D and C down onto E, then a, B prime, C prime would have to lie directly on top of ADE. All right, one last little piece. If AD was ten, AB was 15, BC was 12 and a was 14, then algebraically determine the lengths of DE and AC. Wow. All right, hold on. Wait a second. Well, just for a second, let's draw these two triangles separately. We're going to be doing that a lot in this particular unit. And by the two triangles, I mean triangle ADE. All right? And triangle a, C, B and let's label what they've told me. AD is ten, AB is 15. BC is 12. All right. And I want to figure out the lengths of AC, this guy. And DE, this guy. Now, how do we do this exactly? Well, we do it because we know that this smaller triangle is just a dilation of this larger triangle, right? Or vice versa, the larger triangle is a dilation of this smaller triangle. And let's think about it that way. A lot of people feel more comfortable about it that way. And what would that annihilation factor be? Well, that dilation factor would be this divided by this. 15 divided by ten. All right? So if I wanted to stretch this triangle up so that it was the same as this triangle, I'd have to do it by this ratio of 15 to ten, or if you like. By 1.5. Leave it as 15 divided by ten, though, because the idea, let's say, if we were trying to find DE, let me call it X actually let me just leave it as D why not, right? Then what I could say is I could say, well, let me extend the page a little bit. Whatever I stretch the side ten out by to get the 15 is what I'm going to stretch the DE out to to get 12. So I set up what's called a proportion. And I say, all right. If I take the DE and I divide it by 12, that would have to be the same as the ten. Divided by 15. So DE divided by 12, that dilation factor has to be the same as ten divided by 15. That dilation factor. And we could also flip flop these. It doesn't really matter. Now to solve this, right? You should know the classic technique of cross multiply. I would get 15 times DE is equal to 12 times ten, which is a 120. All right. And then I could just divide both sides by 15. All right, and DE, now I have to think about this a little bit. Is one 20 actually divisible by 15. Why don't I have my calculator? Let me go crap. Yeah, let me go grab a calculator really quick. Ah. Where's my calculator? There it is. Hi little buddy. Thank you for that warning message view, full screen again. So I'll just make it faster otherwise. I'll have to go through the long division of my head. I don't want to do that. All right, and DE is equal to 8. Let me put my calculator up here, so it's not in our way. So we've got our 8. Now we were looking for AC. All right, oh, I never labeled AE is 14. I was going to say it's going to be very hard to find a C on here without AE. That's 14. Let's do exactly the same thing, even with the little red there. We can now say, okay, if I want to take AC and divide it by 14, that's that scale factor. I can use a variety of different things now. I can do the 12 divided by 8. Something like that. Or I could do the 15 divided by ten, either way. But again, just make sure you got it, right? If I do AC divided by 14, I'm really calculating a dilation factor. AC divided by 14 would have to be BC divided by 8. Every side on here has been multiplied by the same thing to get the sides on here. Running out of room. 8 times AC is equal to 14 times 12. Which is one 68. Divided by 8. And AC is 21. A little bit. A little bit crowded there. All right. And that's it. And we're going to get very, very deeply into this type of problem in the next lesson. When we really start to look at what's called similarity. All right? And the idea is very, very simple. If I've got these two triangles, one of which is a dilation of the other, then when I divide two corresponding sides, the number I get will always have to be the same because that's the dilation constant. And whether you're going from the big divided by the small and the small divided by the big, it doesn't really matter as long as you keep that consistent. So here when I do DE divided by 12, DE divided by 12, I'm going smaller, divided by larger, equals ten divided by 15 smaller divided by larger. And again, we'll get into that a lot more in the next lesson. Let's do a wrap up. So in today's lesson, what we saw was a connection between dilations and angles, and the most important property of dilations, the fact that they preserve angles. In other words, when we take a shape and we dilate it, and of course, we're concentrating mostly on triangles. When we take triangles and we dilate them, sure, they're side lengths get larger or smaller depending on the dilation constant, but their angles don't change, which means that the shape, the overall shape of the triangle, remains the same. And that's very important. If I take an image, and I stretch it to be larger or smaller, I don't want it shape to change, I just want it to look like it's at a different scale, so to speak, and that's why we call it a scaling factor. All right, we'll get into this more in the next lesson when we start to look to look formally at what's called similarity. For now, I'd like to thank you for joining me for another common core geometry lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking. And keep solving problems.