Prove that the two circles shown below are similar. Circle X is shown with a center at negative 2, 8 and a radius of 6. Circle Y is shown with a center of 4, 2 and a radius of 3.

Question

please explain step by step

Answer 1

Definition:

Two figures are similar if they have the same shape and the corresponding parts are proportional.

Obviously, the two circles have the same shape. So let's just prove that their corresponding parts are proportional.

There are several ways to do this:

1. In a graph, if you put two circles together in one center and the distance of from one of the circumference to another does not change, then they are similar.
2. I can use a proportion using the radius and the circumference of the two circles: the radius (r) and the circumference (C) which is equal to 2π r

For circle X: radius (r) = 6, Circumference (C) = 12π

For circle Y: radius (r) = 3, Circumference (C) = 6π

Let's see if we have a proportion:

r of circle X / C of circle X = r circle Y / C of circle Y

6/12π = 3/ 6π

reduce both fractions we have:

1/2π = 1/2π.. We have a proportion!

Therefore circle X and circle Y are similar.