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'

Answer 1

I can’t tell which of those three is correct from the question alone — I need the diagram or coordinates (or the intermediate positions A′B′C′D′ and final A″B″C″D″ or the sequence of transformations) to decide.

How you can decide quickly (so you can check and tell me the points if you want):

1. Test for a translation:
- Pick one vertex (say A) and compute the vector from A (pre-image) to A″ (image): v = A″ − A.
- Do the same for another vertex (B): is B″ − B = v as well?
- If the same vector v works for all corresponding vertices, the final transformation is a translation. The sign of the components tells direction:
- x-component > 0 = to the right, < 0 = to the left
- y-component > 0 = up, < 0 = down

2. If it’s not a translation, test for a rotation about B′:
- For each corresponding pair (e.g. A′ → A″), compute the vectors relative to the proposed center B′: u = A′ − B′ and u′ = A″ − B′.
- Check that |u| = |u′| (same distance from B′).
- Check that u′ equals u rotated by 270° about the origin. In coordinates, a 270° counterclockwise rotation maps (x,y) → (y, −x). (Equivalently 270° CCW = 90° clockwise.)
- If that holds for all vertices, the final transformation is the 270° rotation about B′.

If you paste the coordinates or upload the figure (or tell me A′, A″, B′, B″ etc.), I’ll check and give the correct choice.