Question

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

Explanation:

• For two rectangles to be similar, the ratios of their corresponding side lengths must be equal.
• Rectangles can have different side - length ratios. For example, a rectangle with side lengths \(2\) and \(3\) and a rectangle with side lengths \(2\) and \(4\) are not similar. But a rectangle with side lengths \(2\) and \(4\) and a rectangle with side lengths \(3\) and \(6\) (where the ratio of sides is \(1:2\) for both) can be made congruent (after a dilation) and then mapped onto each other using rigid transformations.
• Since rectangles may or may not have equal ratios of corresponding side lengths, a rectangle is sometimes similar to another rectangle. And when the ratios of side lengths are equal (i.e., when they are similar), we can sometimes map one onto the other using only dilations (to make them congruent if they are similar but not congruent) and rigid transformations (translations, rotations, reflections).

Answer:

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