Question

complete the similarity statement.
a circle is sometimes/always/never similar to another circle,
because we sometimes/always/never can map one onto the other
using only similarity transformations.

Explanation:

All circles have the same shape (a round shape defined by a constant radius - to - center distance). The similarity of geometric figures is determined by having the same shape. For two circles with radii \(r_1\) and \(r_2\), we can use a dilation (a type of similarity transformation). The scale factor of the dilation is \(k=\frac{r_2}{r_1}\) (if \(r_2
eq0\) and \(r_1
eq0\)). A dilation is a transformation that changes the size but not the shape of a figure. Since we can always find such a dilation (and translations to map the centers if needed, but translations also preserve shape) to map one circle onto another.

Answer:

A circle is always similar to another circle, because we can always map one onto the other using only similarity transformations (specifically dilations and translations). So the first blank is "always" and the second blank is "always".