## anonymous one year ago Circle B has a center of (-1, 5) and a radius of 4. Circle D has a center of (7, 4) and a radius of 2. Prove that the two circles are similar.

1. anonymous

@ganeshie8

2. ganeshie8

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3. ganeshie8

Circle D can be obtained from circle B by a translation of 8 units to right and 1 units down, followed by a dilation with scale factor of 1/2. Since B can be transformed into D just by translation and dilation, both the circles are similar.

4. UsukiDoll

I remember this proof.... ughhh don't miss it at all >:/

5. ganeshie8

That also proves that all circles are similar!

6. anonymous

so they are similar because it can be dilated to match the size of the other one?

7. ganeshie8

Exactly!

8. anonymous

oh ok so this works for every circle

9. ganeshie8

Yes, coordinates are there just to make the analysis simple. That translation part is not really required. We could simply argue that scaling the circle B by a factor of 1/2 produces the circle D. So both the circles are similar.

10. anonymous

11. ganeshie8

Since any circle can be obtained from any other circle by scaling by a factor of $$\dfrac{r}{r'}$$, we can conclude that "all circles are similar".

12. anonymous

wow i really wish my teacher explained it like you

13. ganeshie8

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14. ganeshie8

im kinda dyslexic today haha @Astrophysics

15. anonymous

so i would put circle b and circle c are similar because you can dilate circle b by 1/2

16. ganeshie8

Also explain the translation part because you're given the coordinates, so i think your teacher expects you to mess with coordinates too

17. Empty

Would it also be considered a proof that all circles are similar since the formula $\pi = \frac{C}{d}$ is a constant? No matter how large the circle is or where it is, $$\pi$$ doesn't depend on it?

18. anonymous

when you say translation what do you mean

19. UsukiDoll

well for an isometric transformation .. translation doesn't do anything to the shape.

20. ganeshie8

I think that can be used, that says circumference scales linearly as diameter is changed

21. ganeshie8

@thehelpyesineed my first two replies in the top cover the complete proof

22. anonymous

okay thanks

23. ganeshie8

you may replace the word "translation" by "move" @thehelpyesineed