Answered

A triangle has vertices at [tex]\( A(-2, -2) \)[/tex], [tex]\( B(-1, 1) \)[/tex], and [tex]\( C(3, 2) \)[/tex]. Which of the following transformations produces an image with vertices [tex]\( A'(2, -2) \)[/tex], [tex]\( B'(-1, -1) \)[/tex], and [tex]\( C'(-2, 3) \)[/tex]?

A. [tex]\((x, y) \rightarrow (x, -y)\)[/tex]

B. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]

C. [tex]\((x, y) \rightarrow (-x, y)\)[/tex]

D. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]



Answer :

GinnyAnswer
Let's carefully analyze each of the given transformations to determine which one produces the desired image of the triangle.

### Original Vertices of the Triangle
- [tex]\( A = (-2, -2) \)[/tex]
- [tex]\( B = (-1, 1) \)[/tex]
- [tex]\( C = (3, 2) \)[/tex]

### Transformed Vertices
- [tex]\( A' = (2, -2) \)[/tex]
- [tex]\( B' = (-1, -1) \)[/tex]
- [tex]\( C' = (-2, 3) \)[/tex]

### Transformation Candidates:

1. Transformation [tex]\( f_1 \)[/tex]: [tex]\((x, y) \rightarrow (x, -y) \)[/tex]
- Applying this transformation to [tex]\( A, B, C \)[/tex]:
- [tex]\( A = (-2, -2) \rightarrow (-2, 2) \)[/tex]
- [tex]\( B = (-1, 1) \rightarrow (-1, -1) \)[/tex]
- [tex]\( C = (3, 2) \rightarrow (3, -2) \)[/tex]
- [tex]\( B' \)[/tex] matches the transformed [tex]\( B \)[/tex], but [tex]\( A' \)[/tex] and [tex]\( C' \)[/tex] do not match.

2. Transformation [tex]\( f_2 \)[/tex]: [tex]\((x, y) \rightarrow (-y, x) \)[/tex]
- Applying this transformation to [tex]\( A, B, C \)[/tex]:
- [tex]\( A = (-2, -2) \rightarrow (2, -2) \)[/tex]
- [tex]\( B = (-1, 1) \rightarrow (-1, -1) \)[/tex]
- [tex]\( C = (3, 2) \rightarrow (-2, 3) \)[/tex]
- All transformed vertices [tex]\( A', B', \)[/tex] and [tex]\( C' \)[/tex] match the required transformed vertices.

3. Transformation [tex]\( f_3 \)[/tex]: [tex]\((x, y) \rightarrow (-x, y) \)[/tex]
- Applying this transformation to [tex]\( A, B, C \)[/tex]:
- [tex]\( A = (-2, -2) \rightarrow (2, -2) \)[/tex]
- [tex]\( B = (-1, 1) \rightarrow (1, 1) \)[/tex]
- [tex]\( C = (3, 2) \rightarrow (-3, 2) \)[/tex]
- [tex]\( A' \)[/tex] matches the transformed [tex]\( A \)[/tex], but [tex]\( B' \)[/tex] and [tex]\( C' \)[/tex] do not match.

4. Transformation [tex]\( f_4 \)[/tex]: [tex]\((x, y) \rightarrow (y, -x) \)[/tex]
- Applying this transformation to [tex]\( A, B, C \)[/tex]:
- [tex]\( A = (-2, -2) \rightarrow (-2, 2) \)[/tex]
- [tex]\( B = (-1, 1) \rightarrow (1, 1) \)[/tex]
- [tex]\( C = (3, 2) \rightarrow (2, -3) \)[/tex]
- None of the transformed vertices match the required transformed vertices.

### Conclusion
After checking each transformation, we see that the transformation [tex]\((x, y) \rightarrow (-y, x) \)[/tex] correctly transforms the vertices of the triangle to [tex]\( A' = (2, -2), B' = (-1, -1), C' = (-2, 3) \)[/tex].

Thus, the correct transformation is:
[tex]\[ \boxed{(x, y) \rightarrow (-y, x)}. \][/tex]

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