Geometry 6.9.4 journal Scenario: Prove That All Circles Are Similar
Instructions
View the video found on page 1 of this Journal activity.
Using the information provided in the video, answer the questions below.
Show your work for all calculations.
The Students' Conjectures:The two students have different methods for proving 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 you 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 you 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?