Question

6.9.4 Journal: Similar circles6.9.4 Journal: Similar Circles

Journal

Geometry Sem 1

Points Possible:20

Name:


Date:



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? (4 points)











Transcript: Similar Circles
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?

Answer 1

Explanation:

Classmate Conjecture

John | All circles are similar if they have the same center and can be dilated or contracted to the same size.

Teresa | All circles are similar if their corresponding radii have a constant ratio.

Evaluate the Conjectures:

2. Yes, it makes intuitive sense that all circles are similar because they all have the same shape and form.

Construct the Circles:

3. [Diagram not provided]

[Diagram not provided]

The hypotenuse of △ABC is the diameter of the smaller circle.

[Diagram not provided]

The hypotenuse of △XYZ is the diameter of the larger circle.

The two triangles, △ABC and △XYZ, are similar because they are both isosceles right triangles with the same angle measures.

The ratio of the lengths of their sides are equal, therefore the ratio of their radii is a constant.

Making a Decision:

10. Both John and Teresa were right as all circles are similar if they have the same center and can be dilated or contracted to the same size (John's method) and also if their corresponding radii have a constant ratio (Teresa's method).

Further Exploration:

11. The circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5 can be found using the Pythagorean theorem to find the diameter of the circle, which is equal to the sum of the lengths of the three sides. The diameter is equal to 5, so the circumference is equal to 2 * pi * (5 / 2) = 5 * pi

Please provide all information when asking a question

Answer 2

ANSWER -

Classmate Conjecture
John
Takes two circles, moves them so they have the same center, then dilates or contracts them until they are the same size.

Teresa
Uses the idea that triangles are similar if all of their corresponding sides have the same ratio. Proves that all circles are similar by showing that if the radii of the two circles have a constant ratio, then the circles are similar.

Evaluate the Conjectures:
2. Intuitively, does it make sense that all circles are similar? Why or why not? (1 point)
Yes, it makes sense that all circles are similar because they have the same basic shape and structure, regardless of their size.

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)
[Diagram not provided]

In your diagram in question 3, draw an isosceles right triangle inscribed inside the smaller circle. Label this triangle ABC. (1 point)
[Diagram not provided]
What do you know about the hypotenuse of △ABC? (2 points)

The hypotenuse of △ABC is equal to the diameter of the smaller circle.
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)
[Diagram not provided]

What do you know about the hypotenuse of △XYZ? (2 points)
The hypotenuse of △XYZ is equal to the diameter of the larger circle.
How does △ABC compare with △XYZ? Explain your reasoning. (2 points)

△ABC and △XYZ are similar triangles because they have the same shape, but different sizes. This is because the hypotenuse of △ABC is equal to the diameter of the smaller circle, and the hypotenuse of △XYZ is equal to the diameter of the larger circle.

Use the fact that △ABC ≈ △XYZ to show that the ratio of the radii is a constant. (2 points)
Since △ABC and △XYZ are similar, we can write the following proportion:
r1 / r2 = AB / XY = BC / YZ

So, the ratio of the radii, r1 / r2, is a constant, which means that all circles with the same center are similar.

Making a Decision
10. Who was right, Teresa or John? (1 point)
Both Teresa and John were right.

Further Exploration:
11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5? (4 points)
[Solution not provided]