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.

- demonchild99

- katieb

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- perl

all circles are similar, but if you want to prove this rigorously.
two figures are similar if you can translate/rotate/scale one figure to another

- perl

http://www.mathsisfun.com/geometry/transformations.html

- perl

If one shape can become another using only Turns, Flips and/or Slides, then the two shapes are called Congruent.
Two shapes are Similar if you need to Resize for one shape to become another (you may also Turn, Flip and/or Slide).

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## More answers

- demonchild99

ok hold on i need a peice of paper.

- demonchild99

ok so before you scale it b would be 7,4 right?

- perl

right, we move the center of circle D to the center of circle B

- demonchild99

when you scale it the 1/2 equals .5 right?.. i mean its a stupid question but you know?

- perl

yes thats correct

- perl

i also graphed the circles so you can see it visually
https://www.desmos.com/calculator/eb0otly6pv

- demonchild99

3.5,2 and cool!

- demonchild99

i mean, thats what i got when i scaled it

- perl

well we are scaling all the points on the circle

- perl

i see what you did, you scaled the center of the circle, but note that the center of the circle is not actually a part of the graph.

- perl

the circle itself is just the circumference or the boundary of the circle

- perl

lets pick a point on circle B , say (1,5)
https://www.desmos.com/calculator/toiyetveam

- perl

now the first transformation will translate or slide the point
to the right 8 units and down one unit

- perl

hmm, ok maybe this idea doesn't work exactly as i envisioned, lets try something different

- demonchild99

lol. ok

- perl

first lets move the circle B to the origin, then scale it by a factor of 1/2

- demonchild99

ok...but wouldnt it still be the origin 7,4

- perl

1. translate center of circle B to the origin (0,0)
2. scale circle B by a factor of 1/2
3. translate center of circle B to center (7,4)

- perl

the origin is the point (0,0) we like to call it the origin of the x y plane

- demonchild99

*facepalm* of course. i knew that

- perl

lets graph it on desmos calculator in steps.
1. Draw circle B
https://www.desmos.com/calculator/q6dcuoxeex
2. Translate circle B to the origin (0,0)
https://www.desmos.com/calculator/l10dl6w4ux
3. scale circle B by a factor of 1/2
https://www.desmos.com/calculator/1vmus35noh
4. translate circle B to center (7,4)
https://www.desmos.com/calculator/3n58wcsjix
now the graph in step 4 is the same as circle D

- demonchild99

ok.. how would i be able to explain this in writing?. i am not very good at turning thing into words.

- perl

now in step 2, when you scaled the center (0,0) by 1/2, thats still (0,0), since a half of zero is 0.
but technically the center is not part of the graph

- perl

translate circle B to origin
(x,y) -> (x+1, y - 5)
scale the moved circle B by 1/2
(x,y) -> (1/2 * x , 1/2 * y)
translate the scaled circle B to (7,4)
(x,y) -> (x +7, y + 4)

- perl

the image of the three transformations is circle D.
since we only used translations and scaling, the two figures are similar

- perl

im not sure how detailed your teacher wants it

- demonchild99

i think i got it now and it should be good! thank you

- perl

briefly in words:
1. slide circle B to origin (0,0)
2. scale the circle by 1/2
3. slide it to point (7,4)

- perl

your welcome :)

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