[G.C.1] Circles #1

Objective

Common Core Text:

• • [G.C.1] Prove that all circles are similar.

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Explanation

We have studied similar triangles, but any type of shapes can have similarity. Circles, even though they have no angles, can similar as well.

Recall,

• • Similarity transformations use

• 1. Translation

  2. Rotation

  3. Reflection

  4. Dilation

Let's take two random circles and see if we can move one onto the other using similarity transformations

We have a circle with radius r and center A, and another circle with radius s and center B.

First, we can translate point A to point B.

Now we just need to make circle A' larger. This needs a dilation. We'll let our center of dilation be A'. But what will our scale factor be? We'll let's think. We want our circle with radius r to transform into a circle with radius s. So what do we multiply r by if we want the answer to be s?

• r x ? = s

Well, you make a fraction with your answer for the numerator and your multiplicand as your denominator. In this case, s is our answer and r is our multiplicand, so the fraction we're looking for is s/r.

Let's see if it works

• r x = s

And that's true. So we'll use