High School: Geometry
1. Prove that all circles are similar.
What goes around comes around. Well, according to Justin Timberlake, anyway.
Students should understand that a circle is a closed curve on a plane. What's special about circles is that all the points on a circle are equidistant from the center. That means circles are defined by two details: position (the center of the circle) and size (the distance from the center to a point on the circle).
Students should know that position isn't a problem when we're talking about similarity or congruence transformations. If we have two congruent figures at different positions, translating them will easily map one on top of the other. But what about size?
Students should know that unlike polygons that have dimensions independent of one another (base and height, for instance), a circle's size depends only on one measurement: the radius r. Sure, we can look at the diameter d or the circumference C or even the area A, but they all depend on r as well. (Now would be a good time to tell them that d = 2r and C = 2πr and A = πr2.)
Since all aspects of a circle's size depend on r, we can change the size of any circle simply by dilating the radius by a constant scale factor. Students should already know that dilations, whether they're expansions or contractions, are similarity transformations. We're changing the size of the circle, but not its shape.
If dilation changes a circle's size and translation can change its position, we can easily map one circle onto another using only those two transformations. Since both dilation and translation are similarity transformations, you can safely tell students that all circles are similar. In fact, we don't even need reflections or rotations to help us out. (Try rotating and reflecting circles and see how they change. Hint: they don't.)
Students can also use two circles and proportions to determine similarity. Two similar shapes will have a constant ratio when corresponding sides are compared. Circles don't have sides, but they do have radii, diameters, and circumferences that we can compare. If students do that, they'll quicky realize that a small circle with radius r and a larger circle with radius r' are similar by a constant scale factor.
Also, we could show in the same way that the area ratio equals the square of the radius ratio.
The circumferences, diameters, and radii of our two circles are all in proportion, which means they're similar. Essentially, this states that circles come in a variety of sizes, but they'll always be the exact same shape: a circle.
Here is a video to refresh your memories about circles.