Identifying rigid transformations

0:00- [Instructor] What we're gonna do in this video is get some practice identifying rigid transformations. Now, what does rigid mean? Well, 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.

0:23So 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 F prime and Figure F. And what we're gonna do is we're gonna look at a bunch of examples of pairs of figures, and we're gonna 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?

0:46So, in this case, how do we go from F to F prime? Well, if we look at these two, they look very similar. In fact, they look identical, these figures. And it looks like they're even oriented in the same way. And so the only difference is they're in different places.

1:02To go from F to F prime, 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. Translation. That's the only rigid transformation at play, to go from F to F prime.

1:24Let's do another example. So what rigid transformation do we need to do to go from Figure K to Figure K prime? 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...

1:42Let 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.

2:06So 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 clockwise, sorry, 90 degrees counterclockwise. So 90 degrees counter, counterclockwise. How do I know this is counterclockwise?

2:30Well, I imagine a clock like this, and this is 12 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.

2:47It's rotating the other way. So that's where I get the counterclockwise from. So this is a 90-degrees counterclockwise rotation. Rotation's just another way of saying a 90-degrees counterclockwise turn. It's going in the opposite direction of what a clock would do.

3:06Let's do another example. 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-degree counterclockwise rotation and a translation. It actually depends what you rotate around.

3:22You 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.

3:39But we'll just focus on the major one here, which is a 90-degrees counterclockwise rotation. I'm not sure if we can argue that these... Well, visually they are in different places here. If we call that first one a translation, I guess we're translating the second one as well.

3:51All right, let's look at this one. It looks like two boots or very simple drawings of boots. What's going on here? 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.

4:08And 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 R prime, it looks like you're doing another 90-degree rotation. But this time we're going in the same direction as the hands of a clock.

4:30So this looks like a 90-degrees 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. All right, so we have Figure C and Figure C prime. How do we go from C to C prime? 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.

5:03If 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. And then that side would actually be over here. It would actually be over here.

5:20And 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 degrees, you could say clockwise or counterclockwise. 'Cause you'd be going all the way around the clock either way.

5:37But 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 could imagine, if there was a mirror right over here, if there was a mirror, you could even imagine like a body of water that can reflect an image.

5:53That's what it looks like. This looks like it's a reflection. And that's the rigid transformation that we actually described this with. This is a reflection. And here we're just trying to identify the rigid transformations, but eventually say, okay, what's it reflected on?

6:07And you would think about 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.

6:25It still has... One way to think about it is a similar relationship between the different sides of these shapes. I'll let you go with that.