To show that circle A is similar to circle B, we can use a series of transformations. Here are the detailed steps:
1. Translate Circle A:
- We start by translating the center of circle A using the rule [tex]\((x + 3, y - 2)\)[/tex].
- The original center of circle A is [tex]\((4, 5)\)[/tex].
- Applying the translation rule, we get the new center:
[tex]\[
(4 + 3, 5 - 2) = (7, 3).
\][/tex]
2. Dilate Circle A:
- Next, we dilate circle A by a scale factor of 3.
- The original radius of circle A is 3.
- After dilation, the new radius is:
[tex]\[
3 \times 3 = 9.
\][/tex]
3. Conclusion:
- After translation, the new center of circle A is [tex]\((7, 3)\)[/tex].
- After dilation, the radius is 9.
So, after translating circle A and then dilating it, we get a circle with center [tex]\((7, 3)\)[/tex] and radius 9. Although circle A has not yet matched the exact position and orientation of circle B, we note that this process preserves similarity because the dilation maintains the radius proportion and the translation changes only the position but not the size or shape of the circle.
Based on these transformations (translation and dilation):
- Circle A and circle B both now have a radius of 9.
- Therefore, these transformations show that circle A can be made similar to circle B through translation and dilation, proving their similarity in terms of their geometric properties.
