Proof: all circles are similar

we're asked to translate and dilate the unit circle to map it on to each circle so this is the unit circle right over here it's centered at 0 comma 0 it has a radius of 1 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 and then dilating it means making it larger so dilating that unit circle would be doing whoops I just translated it 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 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 let me do it for a few more I'm not gonna do it for all of them just do it this is just to give you the idea of what they'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 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 and 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 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