Final answer:
Two circles are similar if they have the same shape, which applies to all circles. Circle C and D are similar because they maintain a proportional relationship between their radii, which is a criteria for similarity. Other options such as area, common chords, or arrangement of centers are not relevant for proving similarity.
Step-by-step explanation:
In mathematics, specifically in the field of geometry, two shapes are similar if they have the same shape, but may differ in size. This concept applies to circles, all of which are similar to each other because they all have the round shape defined by the set of points that are equidistant from a center point.
For circle C with center (2, -2) and radius 2, and circle D with center (0, 3) and radius 6, the criteria for similarity is easily met. What makes two circles similar is that they both have the same shape regardless of their size. In this case, the radii are in a proportional relationship, which is a necessary condition for similarity in geometry. The radius of Circle D is three times that of Circle C, establishing a constant ratio of sizes between them. This directly correlates to the concept that the arc length is directly proportional to the radius of the circular path.
It is important to note that options b, c, and d presented in the question do not provide a basis for proving similarity. Similarity is strictly based on shape and not on equal areas, common chords, or spatial arrangement of centers.