Question

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

Explanation:

Two circles are always similar. All circles have the same shape (they are all circular). The only difference between circles is their size (determined by the radius). A dilation (which changes the size) and rigid transformations (which move the circle without changing its shape or size) can always be used to map one circle onto another. For example, if we have circle \(C_1\) with radius \(r_1\) and circle \(C_2\) with radius \(r_2\), we can first use a translation (a type of rigid transformation) to move the center of \(C_1\) to the center of \(C_2\). Then, we use a dilation with a scale factor of \(\frac{r_2}{r_1}\) (if \(r_2>r_1\)) or \(\frac{r_1}{r_2}\) (if \(r_1 > r_2\)) to make the two circles coincide.

Answer:

A circle is always similar to another circle, because we can always map one onto the other using only dilations and rigid transformations.