Question

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.

Answer 1

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

Answer 2

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

Answer 3

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).

Answer 4

ok hold on i need a peice of paper.

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

yes thats correct

Answer 9

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

Answer 10

3.5,2 and cool!

Answer 11

i mean, thats what i got when i scaled it

Answer 12

well we are scaling all the points on the circle

Answer 13

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.

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

lol. ok

Answer 19

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

Answer 20

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

Answer 21

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)

Answer 22

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

Answer 23

*facepalm* of course. i knew that

Answer 24

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

Answer 25

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

Answer 26

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

Answer 27

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)

Answer 28

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

Answer 29

im not sure how detailed your teacher wants it

Answer 30

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

Answer 31

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)

Answer 32

your welcome :)