In the cat's-head problem, you are doing a transformation that's called dilation. You've seen transformations before -- such as translation, rotation, and reflection. These "rigid transformations" preserve congruence; when you apply one to a figure, it may move around, but it maintains its size and shape.
Dilation (scaling) can change not just the position of the figure, but also its size. There's another non-rigid transformation called a shear that can even change the overall shape of a figure -- the final product, for example, may have different angle measurements than the original.
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