Identifying rigid transformations

[00:00:00]What we're going to do in this video is get some practice identifying rigid transformations. Now, what does rigid mean? Well, if you think if you say something is rigid, you imagine that it's kind of fixed. It's not going to be flexible. And when we're talking about rigid transformations, we're talking about doing things to it that don't fundamentally change its shape. So, if it wasn't rigid, then maybe you wouldn't recognize it as much. It would be more like putty or play-doh. So here I have figure FP prime and figure F. And what we're going to do is we're going to look at a bunch of examples of pairs of figures. And we're going to say, well, what do you need to do? What rigid transformation do you need to do to get from the letter without the prime to the letter with the prime? So in this case, how do we go from F to FP prime? Well, if we look at these two, they look very similar. In fact, they look identical.

[00:00:55]the these figures and it looks like they're even oriented in the same way.

[00:00:59]And so the only difference is they're in different places. To go from f to fprime, I would have to move it in that direction. So the fancy word for moving something without changing its shape in any way because this is a rigid transformation, we would call that a translation.

[00:01:18]Translation.

[00:01:19]That's the only rigid transformation at play to go from f to fprime. Let's do another example.

[00:01:27]So what rigid transformation do we need to do to go from figure K to figure K prime?

[00:01:34]So these are not they don't look like they're in the same orientation. But if we were to take figure K and if we were to rotate it, if we were to take let me look at the corresponding side. So this side looks a lot like this side right over here. And this side looks a lot like this side. I think you see where I'm going with this. This side looks a lot like that side. And then last but not least, this side looks a lot like this side. So if you look at any one of those, for example, this red side right over here, if you were to rotate it 90 degrees to the left like that, and that's 90 degrees clockw, sorry, 90° counterclockwise.

[00:02:23]So 90 degrees counter counterclockwise. How do I know this is counterclockwise?

[00:02:30]Well, I imagine a clock like this. And this is 12, right? right over here. And when you imagine the hands of a clock, the hands of a clock go that way. At the top, they go to the right, and at the bottom, they go to the left. This is going in the opposite direction of that.

[00:02:47]It's rotating the other way. So, that's where I get the counterclockwise from.

[00:02:50]So, this is a 90° counterclockwise rotation. Rotation's just another way of saying a 90 degrees counterclockwise turn. It's going in the opposite direction of of what a clock would do.

[00:03:06]Let's do another example.

[00:03:08]So, and and there's one other thing to point out here in this example. In this first one, you could also argue that maybe maybe you're doing both a 90° counterclockwise rotation and a translation. It actually depends what you rotate around. You could rotate around say this point. You could rotate around a point out here. So there's a lot of different things you could rotate around. But you could argue if you say it rotated around this point, you would get over here and then you would also have to translate to the right. But we'll just focus on the major one here, which is a 90° counterclockwise rotation. I'm not sure if we can argue that these well visually they are in different places here. If we if we call that first one a translation, I guess we're translating the second one as well. All right, let's look at this one.

[00:03:52]It looks like two boots or very simple drawings of boots. What's going on here?

[00:03:58]Well, some of you might be saying, "Look, this side." Whoops. Let me make sure you can see what I'm doing. This side looks a lot like that side, and this side looks a lot like this side, and that side looks a lot like that side. And to orient to go from figure R to figure RP prime, it looks like you're doing another 90° rotation. But this time we're going in the same direction as the hands of a clock. So this looks like a 90° clockwise clockwise rotation. We're going in the same direction as the hands of a clock. All right, let's do one last one of these.

[00:04:45]All right, so we have figure C and figure C prime. How do we go from C to C prime?

[00:04:51]Well, at first you might think there's some type of rotation here, but if you really think about it, if I were to rotate figure C all the way around, it wouldn't look exactly like C prime. If I were to try to rotate it all the way around, it would look something like this. Let me see if I can do this. So, this side, if I were to rotate it all the way around, would look like that.

[00:05:14]And then that side would actually be over here. It would actually be over here. And then you would have something that looks like this. And then that would come out and it would look something like that. That's if you were to rotate it 180 degrees. 180° you could say clockwise or counterclockwise because you're you'd be going all the way around the clock either way. But that's not what's happening here. That's not exactly what that looks like. So here it looks like really a mirror image. You can imagine if there was a mirror right over here.

[00:05:48]If there was a mirror, you can even imagine like a body of water that can reflect an image. That's what it looks like. This looks like it's a reflection.

[00:05:56]And that's the rigid transformation that we actually describe this with. This is a reflection.

[00:06:03]And here we're just trying to identify the rigid transformations. But eventually you say, okay, what's it reflected on? And you would think about some some line that is being reflected across. But here we just have to say this rigid transformation is a reflection. And so once again, all of these, it preserved the fundamental shape of the object. You could recognize it. It didn't bend it. It didn't distort it in any way like that. It still has, one way to think about it, similar relationship between the different the different sides of these shapes. I'll let you go with