Question

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.

Answer 1

1. Move circle with center X, 2 units to the right and 8 units down to produce circle with radius 6 and center at the origin

2. Dilate this circle with a factor of 1/2 (or compress this circle with a factor of 2) to produce a circle with radius 3

3. Now move this circle 4 units to the right and 2 units up to produce circle Y.



Circle X can be transformed into circle Y by translations and a dilation, therefore the two circles are similer

Answer 2

All circles are similar with scale factor R/r where R and r are the radius of the larger and smaller circle respectively.

Answer 3

We can translate the center of circle Y 6 units left and 6 units up making the translated circle Y' with a center at (-2, 8) just like circle X. Then we can dilate the circle Y' by multiplying by a scale factor of 2. Since it has a radius of 3, the dilated circle Y'' will have a radius of 6 matching up with circle X.