:O ok let me just copy it in here:
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 John thinks that circles can be proven similar if 2 of them were placed to share the same center, and one of them were dilated or shrunk to the same size as the other.
Teresa Teresa thinks that all circles can be proven similar if the corresponding sides of circles are proportional, then the circles can be proven similar.
Evaluate the Conjectures:
2. Intuitively, does it make sense that all circles are similar? Why or why not? (1 point)
Yes, because a circle’s circumference/ area is determined by fixed equations, so circles are similar/ proportional.
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)
The hypotenuse of triangle ABC is also the diameter of the small circle.
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)
The hypotenuse of XYZ is also the diameter of the circle.
8. How does △ABC compare with △XYZ? Explain your reasoning. (2 points)
Both triangles are similar to each other by the SAS similarity theorem, and both their hypotenuses are the diameters of their circles.
9. Use the fact that △ABC ≈ △XYZ to show that the ratio of the radii is a constant. (2 points)
R1/R2=AC/XZ
R1*XZ=R2*AC
R1*2*R2=R2*2*R1
They’re equal, hence it’s possible to prove similarity between circles with the ratio of the radii.
Making a Decision
10. Who was right, Teresa or John? (1 point)
I think both of them are correct.
Further Exploration:
11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5? (4 points)