Hey there. This is math 8 unit 2 lesson four looking at dilations on a square grid. And today we're going to be talking about or you talked about in class how to dilate figures on a rectangular grid. You began first of all by estimating some scale factor. So if it said point C is a dilation of point B, that means point C is moving away from point B. With a center of dilation being A, that's our center point and a scale factor of S. Estimate S. What is the scale factor going to be? Well, let's say for example, the length from A to B is this long. We're not going to measure. We just know it's about that long. Just kind of using the eyeball test a little bit here. That same length from here to here is about maybe here to here. That feels about right when I look at it. So we know that for sure if this was one then the length from here to here is about two. A scale factor of two should make it land somewhere about there. But I got a little bit more to go. So if I'm just estimating here, I know I'm for sure not three. Three would extend it out to here someplace. We're not to three. So we're somewhere like two and a little bit more. Maybe you were to say 2 and a4 or something like that. Again, this is just an estimate of what it could be. Okay, it's like a two plus in a sense. So that's estimated scale factor. Now when we look today, we're talking about dilations on a grid. It says to first of all find the dilation of quadrilateral ABC D with center P and a scale factor of two. So what this means is my center is here. All of these points are going to be moving away from P at a scale factor of two. One way to think about it is this because I don't have coordinates yet. The distance from P to A right now is like I'm going up two over one, right? That's my distance up 2 over one. And so what I want to think about is if I do that again with the up two over one, then a ends up being there. I've done that same move twice for a scale factor of two. For D, I'm going down 2 over one. So let's do that again. Down 2 over one. And I have point D right there. When I look at B, I see that B is going up two and over 1 2 3 4 5. So, we'll do that again. We'll double that process. We're going to go up two and over 1 2 3 4 5. Again, all this is from B. How far is it from B? So C is down two and over five. So, we do that again. Down two and over one, two, three, four, five. So I have points here, here, here, and here that are all two times as far away from P as the original. So here's my A prime, my B prime, my C prime, and my D prime. If I was to then, you know, connect the dots, I'd have a rectangle that would look something like this, something like this, something like this. And we notice that this is a rectangle or quadrilateral that has a scale factor of two. And we have to worry about the dilation as well. Look at the length of the original. We have 1 2 3 4 as our length. If this is a scale factor of four, what would you expect to be the length of the new one? Well, four * a scale factor should be 8. Let's take a look. 1 2 3 4 5 6 7 8. So indeed our links are also obviously using the same scale factor of two but the difference with today is now we're using this point this central point uh P to be the point from which everything moves away from it. It's all moving away. And again if you were to think back to previous talks before when we used this the um the this the circle circular grid right we talked about how these points should all be in a line forming that nice ray like so. So, I should be able to line them up. If I was to get my ruler out and do this, they all are going to cross one another. Right now, this was not what you had to do today on this part, but I just wanted to show you once again that that same principle is going to be true. It's a nice way to check to see are you getting it right there. So, the small rectangle has been um with a center of P has been dilated with a scale factor of two and it looks like the red one right there. Okay, let's take a look then at this triangle. It says to find the dilation of triangle QRS with center T and scale factor 2. Notice that for this shape here, they've already provided us with these nice lines, these rays to say our point should land on these dotted lines there. That's the idea. So what I want to do is I'm going to take a look at well how far for scale factor of two. I'm going to look at how far I'm moving from T to Q and T to R and then double that. So T to Q is going up one and over 1 2 3. So instead of going up one over three, I really want to go up two and go over 1 2 3 4 5 6. Or I can think of it as going up one over three and doing it again. up one over three. Okay. To go from R, we're going down one and over 1 2 3 4. Repeat the process because we're factoring 2 * 2 and go down one and over 1 2 3 4. And you notice I'm landing right on that dotted line again. For S, we're going down two over two. We'll do that again and go down two over two and we land right there. So my new triangle here, it's going to be from there to there. So there's those lines. We have here to here for these lines there. And then we have these lines here. Just playing little connect the dots. So that's number two. Our dilation of triangle QRS with center T and a scale factor of two. So the triangle is going by a scale factor of two as is the distance from T growing by two. All right. Now, when we look at the next one, it says, well, let's do dilation of the same triangle with center t and a scale factor of a half, which means we're going to be getting closer. So, this is going to be getting farther away from the P and this gets closer because it's a less than one. So, we can take a look at the same thing again. I erase like my little markings here just to take a look what we want to do. Because going from t to Q was going up one and over 1 2 3. Instead of going up one, I want to go up only a half. And half of three is 1 and a half. So we'd be right there. For r, we went down one and over 1 2 3 4. So again, instead of going down one, half of one is a half. And instead of moving over four, we'll go half of four. Half of four is 1 2. And there we are on the line. And for S, we went down two over two. That would be a simple down one over one. And we'd be right there. And so my triangle is going to be right here. Again, I'm still on those rays, those lines. Everything looks good. And now I have these copies of the original with the scale factor of two, right? This is a times two and a scale factor of a half. And that was what was happening there. What's really cool is that we're adding this idea of that we're going from here. We have this grid, but we're going to move now to a coordinate grid, which is going to allow us to have a real concise way to name points. So, instead of just saying it's out there or it's around there, using a coordinate grid allows us to really make put a name to these dots. Where do they live? Where's where's their home at? Okay. So, the next thing you did is you did a card sort activity in class. I do want to take a moment to go over this um just because it's good to see it again and it will come up again and again. So you had some cards and with those cards you had some some rules in a sense or some some matches. So it was like you had a shape like this and then you're supposed to do a dilate the trapezoid using center whatever and a scale factor and it's supposed to move around. Okay. And you weren't you then were tr supposed to find some coordinates for what that could look like in the end. what would that look like when you went through it? So to do this here, um it's it's important to think of a couple things. One, you have to look at where your center is going to be. First of all, so my center is -1, -2. This is my center, which means that all the dilations are being are moving from this point. And in this case here, our scale factor is 3 over2. So I'm moving away from that center. and a scale factor 3 over two. So let's take a look at point A first of all. So point A when you begin with it, point A and let's we're going to look it like this. I like to break it up this way. I have my x value and my y value, right? So a currently is 1 2 3 4 units away from the center in terms of the x value. I'm going to take that and I'm going to multiply it by the scale factor 3 over 2. And that becomes 4 * 3 is 12 / 2 is 6. So I want this point to move 6 units away from the center. So that would be 1 2 3 4 5 6 right there. Okay, that's all I'm having to do there because it's on the same line. So I can leave it there like that. Do I need to see a yvalue? Not really, because the y is no distance is no distance, right? From the y to y, it's just a zero times.32. Nothing moves. So, it stays exactly where it's at. That's my a prime. Okay, that's not too bad. Now, when you took a look at something like, okay, let's look at our next letter. If I'm looking at, let's say, C, C has coordinates right now. The x coordinate, the x value, sorry, is how far away from b? If b is here at -1 and c is at -2, what's the distance there? We would say it's one away, right? Okay. If we take that 1 and multiply it by a scale factor of 3 over2, we end up with 3 over2 or we could write that down as 1 and 1/2. So, I'm going to plot a point that's 1 and a half units away from the B. The B part. Get there in a second. For the vertical part, we notice that we're at 1 2 3 away vertically from B. 1 2 3. I'll multiply that by 3 over2. And when we do that, we have 9 over two or 4 and a half away. Okay. So, let's go back to this point right here. What that's going to mean is sorry for the x value I want to move 1 and a half away from b one and a half I'm going to be there and for my y value I'm going to move away 4 and 1/2 up. So I'm going to go from the half one 2 3 4 and a half up right there. Okay. So, this is going to be where right now I believe our C prime value is going to be. It might be a good idea just to stop and check a couple things before I get to point D. Remember, these should all form a nice line, shouldn't they? If I was to continue this ray forward, everything should connect. And when we double check that, that's connecting. And of course, this one, because it's on a line, that's connecting. No problem. When I look at D, my answer for D should also end up being on this line. That's where it should land at some point there. So, let's take a look at D. D has a distance away from the B value of 1 2 3. It's three away. And in terms of the Y value, again, it's also 1 2 3 away. I'm going to multiply that by the scale factor. multiply both by the scale factor. And when you multiply that by the scale factor, 3 * 3 over2 is 9 / 2. And we're at 4 1/2. And 3 * 3 over2 is 9 /2, which is 4 and 1/2. So I get the same number both times at 4 and 1/2. 4 and 1/2. So what I'll do is I'll move from this point 4 and 1/2 to the left, right? 4 and 1/2 this way. And then four and 1/2 up. So, we're going to go 1 2 3 4 and a half. And then go 1 2 3 4 and a half. And look at that. We land right there on that line D prime. And so if I was to map this out, I would have a line here going across there sharing that point. We would have this line there. We'd be able to play connect the dots across the top. and then I can connect this line right there. Okay, so that is what you end up drawing and how you do that. Now, in terms of what you had as a solution, one of your solution choices would have looked like this and you would have had -7, -2. Here's our -1, -2. That's our point that stayed the same. and then your -2 and 1/2 and then positive 2 and 1/2, - 5 and 1/2 and positive 2 and 1/2. This was probably one of the more complicated ones that you had from this little activity. So, I wanted to show that to you real quick. And that's how you did that one there. Okay, that was number one. Number two was another good one. Same idea, right? So, here's number two. And it says that for number two you have to dilate the trapezoid using a center of again -3 comma -4. So that center in this case -3 comma pos4 sorry3 pos4 is up here. That's our center point right there. And we're going to have a scale factor of a half. So that means it's going to be getting smaller one. the shape will get smaller and it's also going to become closer to our center. Those are two things we can tell by that scale factor. It's going to be smaller and closer to our center. That's the first thing. Now, I can also while we're thinking about it, just kind of easy to do. Everything is going to be connected on this line. So, the points that I find are going to be on this line somewhere. Okay? That's just the rule that we live by. So, it's going to be on this line. So this new shape I'm going to draw is going to fit somewhere in that space. Okay, let's take a look at what what we're given. First of all, we're given some coordinates here to start with. And so let's start with point. We'll start with point D. So D currently in terms of the distance away from the point D is one away from the X value. It's one distance away. in terms of the vertical or the y, we're 1 2 3 down. That's our distance from our center point. If I multiply that by the scale factor, multiply that by the scale factor, then I find that I need to move half a click away from the center point and over here 3 and 1/2 away down. So, what that means in our origin land is we're going to go half over here. and three and a half means 3 over two is one and a half right there. So that's where point D prime is going to be. The same thing is going to be true or very similar for C because C is also if we look at it, it's one away and three down. So it's going to end up being right across at this point right there. When we looked at A, A comes up this line here. And so a is actually two away. So for a being two away on the x value, the y value from here is 1 2 3 4 5 6. So I multiply 2 * a half and you get 1. I multiply six by a half and we get three. So that tells me for a I'm going to go over one and I'm going to go down three. One, two, three. And I'm going to land right there. And this would be point A prime. B is the same thing as A except it's over to this side. And so B would be here for B prime. And so that trapezoid would end up landing here, here, here, and here. And that's how that's going to work. It is smaller and it's also closer to our center. Okay. So that was number one and two. and how that worked out. And so you might have had a card that matched something like this. -4 comma 1 -2 comma 1 and our 2 and 1/2 2 and 1/2G3 and 1/2 2 and 1/2 like so. All right. So that's how you do that process there. But I want to make sure you saw that. Okay. Some of the other ones weren't quite as as complex because they were just circles. You're used to those here. If this is our center and has a scale factor of two, that means that instead of two, we double it. We go to four. We go from two to four. We go from negative2 to neg4 and negative -2 to neg4. And then you end up just drawing that circle like so. And you have those kind of points for that one. So it's pretty straightforward when it's just coming from the origin. And we're just moving out a nice simple number there. That was number two. When you looked at number four, card four was the same idea. We're already going from the origin again. You're moving out two, but we're going to cut that in half. So instead of two, we go 2 * a half equals 1. We have points at 1 1. And our circle then just goes inside of there. And those become your new points for that one there. And then your triangle. A good one as well. When you dilate from the origin using a scale factor of two, we can look and say, well, in this case, I looked at units. I'm going 1 2 3 4 5 units. So 5 * 2 is 10. So 5 6 7 8 9 10. I would just carry this line out to this point which is five technically which makes sense because it's 2 and 1/2. And then for my height I was currently at 1 2 3. And so I'm going to go up 1 2 3 4 5 6 and put that one right there. And because from the origin, I just have to connect the dots and I'm good for that one with points being here, here, and here. I'm dialing that triangle using the scale of two. And that's the way that goes. Okay. The orange is not going to move because if you take that point 0 and 0 and multiply by the scale factor of two, you still end up with 0 0. It stays exactly where it is. And finally, the last one you looked at was here. D triangle using center -4, -3. That means the center is right there. That's our center point there. And we're going to do a scale factor of 1.5. So 1.5. So what you took a look at here is how far away was that triangle from from the starting point. Again, if that's my my starting point there, I could think about everything needs to fall on this line, right? Because that's my line I'm working with. That's going to be there. So, when you look at your first point, my first point here for let's call that A, A is at in terms of how far away it is, it's 1 2 3 4 units away from the center already. And we're doing a scale factor of 1.5. So 4 * 1.5 4 * 1.5 gets you 6. So our a ends up landing right there for a prime. This one was already six away. So 6 * 1.5 equals 9. So we have 6 7 8 9. Our next point lands here. And then if we were just doing scale factors, 2 * 2, sorry, 2 * a 1.5. We're going to go across to three. one, two, three, and draw the line there. So, our triangle is going to be here, here, and here. All right. So, that was the part of the card sort that you did by yourself. I know I took a little more time than that. If you fast forward through it, that's okay, but I wanted to make sure you saw how to do that because it may be something you didn't have a lot of time to talk about in class. Let's take a look at the are you ready for more real quick before we look at tonight's homework. In terms of the are you ready for more you have a circle here and it says what happens to center dilation is a point in a circle um using center 0 as our point let's do a scale factor of 1.5. Okay. So here's what I did to do this. If I'm at 1.5 I know I do have some points here because of just the length of it. I have a point here at 4a 4. I have a point at 0 comma 8. And so I use those points right there. They're just easy to easy to work with. So if I had a scale a point here at four, comma four and that's my origin, right? I'm four away and I'm four away. Four * a 1.5 is going to give me six. And my other four, that's my x value, my y value in terms of how far from the origin is also 1.5, which gives me six. So this point here would move to six, comma 6, right there. So that's where that point moves to. And because it's a circle, it's going to also come down here to 6, -6. When I took this point, we can call that point X. Now we'll call it point B. When I took point B, point B happened to be 8 away from the x value and zero away from the y value. Multiply by 1.5 * 1.5. This is still zero. And 8 * 1.5 is 12. So my other point lands over here at 12 comma 0. So in terms of a circle, what we're going to have is we're going to have a circle that kind of comes up this direction, hits there. It's going to come back around here and get our 12. It's going to come down and get our 6, -6, and back up there. Can you use these points to find other points on there? You sure can. And I'll let you do that on your own. And so moving to the next page, we are going to talk about this just the summary real quick this lesson. Again, sorry with the length on those cards, but just thought it'd be helpful for you to understand that. So the point is that rather than using a ruler to measure the distance between points, we can count grid units from time to time. We can see how far it is, click, click, click, to go from one to the other without having to get a ruler out because a ruler doesn't always work very nicely. And sometimes the coordinate grid gives us a more convenient way to name the points and that's a helpful way for us to identify different elements of of uh how these things are moving around. Okay, let's pause there for a moment and work on your homework and then come back and press play and let's see how you did. [Music] Okay, so here is the homework for tonight. Okay, rule number one. It starts with triangle ABC is dilated using D as a center dilation with scale factor 2. Okay, we're dilating things with D is a center. We're doing a scale factor 2. That's great. Question. The image of A prime, B prime, C prime. Claire says the two triangles are congruent. Congruent means they are going to be the same. Correct? Here's the question. She says that's because their angle measures are the same. They look similar, right? They don't look that different from the angle measurements. I'm not going to bother measuring them. But angle measurements never change. That's true with a triangle or with shapes. The angles stay the same. But if it's congruent, are the sides the same? That's the big question. Are the sides the same here? Well, we would probably say nope, they are not. That's a pretty long side compared to that shorty side right there. So, we would say no, these are not congruent triangles. Don't even worry about the dilation part because even if it was dilated correctly, which it doesn't look like it was either, the sides are not the same. So, it's not a congruent triangle. Okay. On number two says on graph paper here, sketch the image of quadratal PQRS under the following dilations. So let's look at the first one. The first one says dilation centered at R with a scale factor of two. Okay. So let's look at the first one. We'll do each one in a different color. How about that? Okay. Let's start this first one with um we'll go with a pink. So scale factor two with R being our center point. So the S is already one two away. We're going to double that. Make it go 34 away for our point for S prime. Okay, P is going to be currently at 1 2 and over and then 1 2 3 up. So we'll do that again. 1 2 over 1 2 3 up. So that's where my P prime is going to be. And Q is at 1 2 3 4 over one up. We'll do it again. 1 2 3 4 over and one up. And that's where Q prime is going to be. And so when I connect the dots there, we have a line here. This line continues along that line. This line also continues along that line. And then this one's a new line connecting here to here. So that's our first one and what it should look like. For our second one, the dilation centered at at O right here with a scale factor of a half. So what I'm doing this time is I'm seeing how far away is the initial one from the Q from the O. Q currently is 1 2 3 away. Half of three. What's half of three? So 3 * a half is 1 and 1/2 or 3 over two. So Q is going to be at 1 and a half away. There's my first point for the Q. P is currently over two and up one two three four five. So we're up five. We're up five and we're over two. So instead of going over two, we're going to do half of that. So half of over two is going to be over one. And half of going up five, all right, sorry, half of up five is going to be uh two and a half, right? So 2.5, two and a half. So, we're going to go over one and we're going to go up 1 2 and a half about there. R is currently 1 2 3 4 over and up two. Half of four is 1 2. Half of two is 1. So, that's where R is going to be. And then for the S value, we're over 1 2 3 4 5 6. What's half of 6? 6 * a half is 3. So, we'll go over three, but then we're up one, two, three, four. So, we're going to go up two because half of four is two. And we'll put it right there. Wait, nope. Sorry. Over here. My mistake. Sorry about that. Put it right there. Forgot about my three and two. So, it's going to be here to here to here to here. And that's that one there. Now for our final one, the dilation of S, dilation centered at S, which is right here. Here's our center point. Scale factor of a half. So it's the same shape but scaled at a half. So in this case, we went over or we went up one and over 1 2 3 4. So we're going to go up a half and over half of that, which is two. It should be on the line. For this one, we went down two over two. So now we'll just go down one over one, right? We're cutting it in half. And this guy, we went down one and over one 2 3 4 5 6. So we'll go down half and over three. 1 2 3 because half of six is three. So that shape fits there, there, and there. And this shape should be the same as the green one because it's also a scaled factor of a half, but it's just different location because of what it's centered around. So that's what it looks like for number two. The three different shapes you're going to create there. All right, coming to the end here. It gets faster from this point, I promise. Okay, so here we have some lines. We're marking to see what the angle measurements are. We have a vertical angle here, and the vertical angle becomes 35°. What's left is supplementary. So we have 180 minus 35 and what's left is 145 degrees here. And that would also make it 145° there. I'm going to let you do that one on your own because it's following the same process and you can figure out what that's going to be. You start with a 27 and work your way around. For number four, describe the sequence of reflections, rotations, translations to get from there to there. this shape here. If we were to get it to that shape, we're gonna have to probably rotate it around probably about 90 degrees at least clockwise, right? And then we'd have to do some sort of translation to be able to align that up how that's going to be. You write down what the what you want them to do, but you probably to rotate it and then translate it to the right and to make that work the right way. Okay. And then number five, it says point B has coordinates -2,5. So -2 and then 1 2 3 4 5. So it's right about there. After a translation of four units down, we're going down this way. Four and a reflection across the x axis. We reflect. We're going to land over here. Right there. And a transition up six. So four, five, six. plus six. Where do we end up at? Well, we end up right here. And right there is going to be at 2, three, negative three, sorry. So, I got to get my positive negative right. And that's your solution there. Hope that helps you out a little bit today. It's a good little lesson. Took a little bit of time to work that out, but we'll keep using it. Have a good day.