Question

Which composition of similarity transformations maps polygon ABCD to polygon A'B'C'D'?

a dilation with a scale factor of One-fourth and then a rotation
a dilation with a scale factor of One-fourth and then a translation
a dilation with a scale factor of 4 and then a rotation
a dilation with a scale factor of 4 and then a translation

Answer 1

I don’t have the diagram or coordinates, so I can’t pick one of the four choices directly. Tell me the picture or the coordinates, or use the quick test below to decide which composition is correct.

Steps to decide:

1. Find the scale factor.
- Measure any corresponding side lengths (for example AB and A′B′).
- Compute k = |A′B′| / |AB|.
- If k = 1/4 the correct answer must be one of the first two choices (a shrink). If k = 4 it must be one of the last two (an enlargement).

2. Decide whether a rotation is needed or only a translation after the dilation.
- Compute a direction vector for one side before and after, e.g. v = B − A and v′ = B′ − A′.
- Find the angle between v and v′ (or compare slopes).
- If v′ is parallel to v (angle 0° or 180°) then the figure has the same orientation and only a translation (after the dilation) is needed → choose the “then a translation” option.
- If v′ is not parallel to v (angle ≠ 0°,180°) then the figure is rotated relative to the original → choose the “then a rotation” option.

If you give me the diagram or the coordinates of A,B,C,D and A′,B′,C′,D′ I’ll compute k and the correct composition and show the work.