Similarity Transformation Homework

Question

Do the compositions T<4, –2> ◦ D2 and D2 ◦ T<2, –1> describe the same similarity transformation? Explain.

Answer 1

Yes, they will yield the same transformation.

Imagine you had one starting point (x, y).

If you do the first transformation T<4, -2> o D2, you would do the dilation first to yield (2x, 2y) and then the translation to get (2x + 4, 2y - 2). (When you do a composition of transformations, you start with the right and then you do the left.) (Dilation means you multiply everything by the factor of 2. Translation means you move right 4 and down 2.)

If you do the second transformation D2 o T<2, -1>, you would do the translation first to yield (x + 2, y - 1) and then the dilation to get (2x + 4, 2y - 2).

Notice that in both cases (x, y) became (2x + 4, 2y - 2). Therefore these are the same transformations.