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]\(A^{\prime}(2, -2)\)[/tex], [tex]\(B^{\prime}(-1, -1)\)[/tex], and [tex]\(C^{\prime}(-2, 3)\)[/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 examine each of the given transformations to determine which one transforms the vertices of the original triangle [tex]\(A(-2, -2), B(-1, 1), C(3, 2)\)[/tex] to the desired vertices [tex]\(A'(2, -2), B'(-1, -1), C'(-2, 3)\)[/tex].

1. Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[ A'(-2, -2) \rightarrow (-2, -(-2)) = (-2, 2) \][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[ B'(-1, 1) \rightarrow (-1, -1) \][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[ C'(3, 2) \rightarrow (3, -2) \][/tex]

This transformation gives vertices [tex]\((-2, 2), (-1, -1), (3, -2)\)[/tex], which do not match the desired vertices.

2. Transformation: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[ A'(-2, -2) \rightarrow (-(-2), -2) = (2, -2) \][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[ B'(-1, 1) \rightarrow (-1, -1) \][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[ C'(3, 2) \rightarrow (-2, 3) \][/tex]

This transformation gives vertices [tex]\((2, -2), (-1, -1), (-2, 3)\)[/tex], which match the desired vertices.

3. Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[ A'(-2, -2) \rightarrow (-(-2), -2) = (2, -2) \][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[ B'(-1, 1) \rightarrow (-(-1), 1) = (1, 1) \][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[ C'(3, 2) \rightarrow (-3, 2) \][/tex]

This transformation gives vertices [tex]\((2, -2), (1, 1), (-3, 2)\)[/tex], which do not match the desired vertices.

4. Transformation: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[ A'(-2, -2) \rightarrow (-2, 2) \][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[ B'(-1, 1) \rightarrow (1, 1) \][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[ C'(3, 2) \rightarrow (2, -3) \][/tex]

This transformation gives vertices [tex]\((-2, 2), (1, 1), (2, -3)\)[/tex], which do not match the desired vertices.

From the above transformations, we see that the transformation [tex]\((x, y) \rightarrow (-y, x)\)[/tex] produces the required image with vertices [tex]\(A'(2, -2), B'(-1, -1), C'(-2, 3)\)[/tex].

Thus, the transformation that produces the desired image is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]