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## High school geometry

### Unit 2: Lesson 4

Symmetry# Intro to rotational symmetry

CCSS.Math:

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

## 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.