Question

AB¯¯¯¯¯¯¯¯
is rotated 120° clockwise about B. Then AB¯¯¯¯¯¯¯¯
is rotated 45° counterclockwise about A. What is the image of A as a composition of transformations?

A. (r(120°, B) ∘ r(–45°, A))(A)
B. (r(–45°, A) ∘ r(120°, B))(A)
C. (r(–120°, B) ∘ r(45°, A))(A)
D. (r(45°, A) ∘ r(–120°, B))(A)

Answer 1

The problem involves applying transformations (rotations) to point \( A \) in a specific order. We are given that the segment \( \overline{AB} \) is first rotated 120° clockwise about point \( B \) and then rotated 45° counterclockwise about point \( A \).

To express this as a composition of transformations, we follow these steps:

1. The first transformation is the rotation 120° clockwise about point \( B \), denoted as \( r(120°, B) \).
2. The second transformation is a rotation of 45° counterclockwise about point \( A \), denoted as \( r(-45°, A) \).

When we apply transformations sequentially, we start with the innermost operation. So, we first apply \( r(120°, B) \) to point \( A \), and then apply \( r(-45°, A) \) to the result of that operation.

This means that the correct composition of transformations should be:

\[
(r(-45°, A) \circ r(120°, B))(A)
\]

Thus, the answer is:

**B. \( (r(-45°, A) \circ r(120°, B))(A) \)**