Can circles be proven similar with ratio of radii? If so, how does it work?
Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
since circles are similar, the ratio of the circumference and diameter has to be a fixed number. But then it turned out that, unlike with squares for example, that this number is not so "easy". And as Archimedes in very early ages (and others later on), started investigating this, it turned out that this ratio had a lot of decimals. And so it made sense to give this number some abbreviation. The pi symbol came much later though, somewhere in medieval times, give and take few centuries. It was proven transcendental in eighteenth century
ooooo thnkx. But I still have one question :/ I'm a little unsure about my selection for this problem and this assignment in general :P. Would you mind explaining parts of it to me?
:O ok let me just copy it in here:
Scenario: Prove That All Circles Are Similar
• 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)
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)
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.
11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5? (4 points)
I think you did good job on this assignment
And plus you explained your answers perfectly