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Intro to rotational symmetry

Sal checks whether various figure are symmetrical under a 180 degrees rotation. Created by Sal Khan.

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Video transcript

We have two copies of six different figures right over here. And I want to think about which of these figures are going to be unchanged if I were to rotate it 180 degrees? So let's do two examples of that. So I have two copies of this square. If I were to take one of these copies and rotate it 180 degrees. So let me show you what that looks like. And we're going to rotate around its center 180 degrees. So we're going to rotate around the center. So this is it. So we're rotating it. That's rotated 90 degrees. And then we've rotated 180 degrees. And notice the figure looks exactly the same. This one, the square is unchanged by a 180-degree rotation. Now what about this trapezoid right over here? Let's think about what happens when it's rotated by 180 degrees. So that is 90 degrees and 180 degrees. So this has now been changed. Now I have the short side. I have my base is short and my top is long. Before my base was long and my top was short. So when I rotate it 180 degrees I didn't get to the exact same figure. I have essentially an upside down version of it. So what I want you to do for the rest of these, is pause the video and think about which of these will be unchanged and which of them will be changed when you rotate by 180 degrees. So let's look at this star thing. And one way that my brain visualizes it, is imagine the center. That's what we're rotating around. And then if you rotate 180 degrees. Imagine any point. Say this point, relative to the center. If you were to rotate it 90 degrees, you would get over here. And then if rotate it 180 degrees, you go over here. You go the opposite side of the center from where it is. So from that point, to the center, you keep going that same distance. You'll end up right over there. So this one looks like it won't be changed. But let's verify it. So we're going to rotate 90 degrees. And then we have 180 degrees. It is unchanged. Now let's look at this parallelogram right over here. So its center, if we think about its center where my cursor is right now-- Think about this point. The distance between that point and the center, if we were to keep going that same distance again, you would get to that point. Likewise, the distance between this point and the center, if we were to go that same distance again, you would to get to that point. So it seems like that point would end up there. That point would end up there. And vice versa. So I think this one will be unchanged by rotation. So let's verify it. So you go 90 degrees. And then you go 180 degrees. Or I should say, it will be unchanged by rotation of 180 degrees around its center. We got the same figure. Now let's think about this triangle. So if you think about the center of the figure. Let's say the center of the figure is right around here. If you take this point, go to the center of the figure, and then go that distance again, you end up in a place where there's no point right now. So that point is going to end up there. This point is going to end up there. This point is going to end up here. So you're not going to have the same figure anymore. And so we can rotate it to verify. So that's rotated 90 degrees. And then that's rotated 180 degrees. So we've kind of turned this thing on its side. It is not the same thing. Now let's think about this figure right over here. Well this figure, if you rotate it 180 degrees, this point is now going to be down here. And this point is going to be up here. So you're going to make, essentially it's going to be an upside down version of the same kite. And we can view that. We can visualize that now. So it's going to be different. But let's just show it. This is 90 degrees. And now this is 180 degrees. If it was actually symmetric about the horizontal axis, then we would have a different scenario. We would have a different scenario with this thing right over here. If it was some type of a parallelogram, or a rhombus, or something like that, then this could have been more interesting. But in this situation, if it was just a more symmetrical diamond shape, then this rotation would not have affected it. But this one clearly did.