The video begins with a young woman talking in front of a blank screen. Audio:

I'm Teresa. My friend John and I need to prove that all circles are similar. It seems obvious, right?! Of course they're similar, they’re all circles!

[Many circles of different sizes and colors pop up onto the screen.]

Um, this is making me a little dizzy.

[The circles disappear.]

But we do need to prove that all circles are mathematically similar.

Here’s the way John looks at it: Remember what we learned about similar triangles?

[Two triangles appear on the screen. One is small and the other is large.]

We can take one triangle, and move it on top of another triangle.

[The small triangle is placed on top of the large triangle.]

Then, we dilate it to show that they are similar. Like that.

[The small triangle is dilated to the size of the large triangle.]

John says we can do the same thing with circles.

[Two circles appear on the screen. One is small and the other is large.]

Take any two circles, and move them so that they have the same center.

[The small circle is moved on top of the large circle.]

Then, you can dilate or contract the circles until they are the same size.

[The small circle is dilated to the size of the large circle.]

Taa-daa! The circles are similar. I have another way to prove it.

[Two triangles appear on the screen. The small triangle has sides of length 2, 2, and 3 and the large triangle has sides of length 6, 6, and 9.]

We also know that triangles are similar if all of their corresponding sides have the same ratio.

[The corresponding sides of the triangle are highlighted. On-screen text: 2 over 6 equals 2 over 6 which equals 3 over 9 which equals 1 over 3 Similar!]

Well, the same idea should also work with circles.

[Two circles appear on the screen. One circle is small and the other circle is large.]

If the corresponding parts of two circles have the same ratio, then the circles must be similar. And lucky for us, everything about a circle can be described with its radius!

[Beneath the small circle is written "equals r sub 1." Inside the large circle is written "equals r sub 2." On-screen text: Diameter equals 2r, Circumference equals 2pi r, and Area equals pi r squared.]

So, if the radii of these circles have a constant ratio, then the circles are similar.

[On-screen text: If r sub 1 over r sub 2 equals k, a constant, then the circles are similar.]

What's more, I think I can prove all this by using inscribed triangles. But I need your help.

[A triangle is inscribed in each of the circles using the diameter of the circles as their bases.]

Can we actually use inscribed right triangles to show that all circles are similar?

1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)
Classmate Conjecture
John

Teresa

Evaluate the Conjectures:
2. Intuitively, does it make sense that all circles are similar? Why or why not? (1 point)

Construct the Circles:
3. Draw two circles with the same center. Label the radius of the smaller circle r1 and the radius of the larger circle r2. Use the diagram you have drawn for questions 3 – 10. (2 points)

4. In your diagram in question 3, draw an isosceles right triangle inscribed inside the smaller circle. Label this triangle ABC. (1 point)

5. What do know about the hypotenuse of △ABC? (2 points)

6. In your diagram in question 3, extend the hypotenuse of △ABC so that it creates the hypotenuse of a right triangle inscribed in the larger circle. Add point Y to the larger circle so it is equidistant from X and Z. Then complete isosceles triangle XYZ. (1 point)

7. What do know about the hypotenuse of △XYZ? (2 points)

8. How does △ABC compare with △XYZ? Explain your reasoning. (2 points)

9. Use the fact that △ABC ≈ △XYZ to show that the ratio of the radii is a constant. (2 points)

Making a Decision

10. Who was right, Teresa or John? (1 point)

Further Exploration:

11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5? (4 points)