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Studying for a test? Prepare with these 14 lessons on Circles.

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# Proof: all circles are similar

Video transcript

We're asked to translate and dilate the
unit circle to map it onto each circle. This is the unit circle right over here;
it's centered at (0,0), it has a radius of one. That's why we
call it a unit circle. When they say translate, they say move it
around. So that would be a translation of it.
Then dilating it means making it larger. So dilating that unit circle would be
doing something like that. So we're going to translate and dilate this
unit circle to map it onto each circle. So, for example, I can translate it so
that the center is translated to the center of that magenta circle, and
then I can dilate it so that it has been mapped on to that
larger magenta circle. I can do that for a few more. I'm not
going to do it for all of them. This is just to give you an idea of what
we're talking about. So now I'm translating the center of
my -- it's no longer a unit circle -- I'm translating the center of my circle to
the center of the purple circle and now I'm going to dilate it so it has
the same radius. And notice, I can map it. And so if you can map one shape to another
through translation and dilation, then the things are, by definition they
are going to be similar. So this is really just an exercise in
seeing that all circles are similar. If you just take any circle and you make
it have the same center as another circle then you can just scale it up or down to match the circle that you moved it
to the center of. So there you have it. Hopefully this gives you a sense that
all circles are similar.