Question

What is the final transformation in the composition of transformations that maps pre-image ABCD to image A"B'C"D"?

a translation down and to the right
a translation up and to the right
a 270° rotation about point B'
a 180° rotation about point B

Answer 1

I don't have the diagram or coordinates, so I can't tell which choice is correct yet. Give me the picture or the coordinates of A, B, C, D and of A", B", C", D" (or a clear description of the intermediate steps) and I'll pick the correct final transformation.

If you want to decide it yourself, use these tests:

- Test for a translation: compute the vector from each original point to its image (for example v = A" − A'). If the same vector v works for all corresponding points, the transformation is a translation. If v.x > 0 and v.y < 0 it is "down and to the right"; if v.x > 0 and v.y > 0 it is "up and to the right".

- Test for a 270° rotation about point B': check that B' is fixed (B" = B') and for any other point P', the vector from B' to P" equals the 270° rotation of the vector from B' to P'. In coordinates, if u = P' − B' = (x,y) then the rotated vector should be (y, −x); so P" − B' should equal (y, −x).

- Test for a 180° rotation about point B: check that B is fixed (B" = B) and for any point P (or P' depending which stage is being rotated about B), the vector from B to P" equals −(vector from B to P). Equivalently, B should be the midpoint of P and P".

Send the coordinates or an image and I'll determine which of the four is the final transformation.