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 pick the correct choice yet. Please either upload the picture or give coordinates for A, B, C, D and A″, B″, C″, D″ (or at least A′ and A″ and B′).

How to tell which of the four it is (quick tests you can do):

1. Test for a translation:
- Compute the vector v = A″ − A (componentwise). If B″ − B = v and C″ − C = v and D″ − D = v, the transformation is a translation.
- If v has x > 0 and y < 0 it is “down and to the right.” If x > 0 and y > 0 it is “up and to the right.”

2. Test for a rotation about B′:
- For each original point P, form the vector from B′: u = P − B′ and for the image point P″ form u′ = P″ − B′.
- If |u′| = |u| for all points and u′ = −u (i.e. opposite direction), that is a 180° rotation about B′.
- If |u′| = |u| for all points and u′ is u rotated by 270° (counterclockwise) — algebraically, if u = (x,y) then u′ = (y, −x) — that is a 270° rotation about B′.

Give me the image or coordinates and I will identify which of the four choices is the final transformation.