Proof: all circles are similar

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.