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To determine which two options describe the rotation of the triangle with vertices [tex]\(X(0,0)\)[/tex], [tex]\(Y(0,-2)\)[/tex], and [tex]\(Z(-2,0)\)[/tex], resulting in the image triangle with vertices [tex]\(X'(0,0)\)[/tex], [tex]\(Y'(2,0)\)[/tex], and [tex]\(Z'(2,-2)\)[/tex], we need to analyze the transformations that would map the original points to their corresponding new locations. Let's go through the possible transformations step by step:
1. 90 Degree Rotation Counterclockwise ([tex]\(R_{0,90^\circ}\)[/tex]):
- For a 90-degree rotation counterclockwise about the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- Applying this to the vertices:
[tex]\[ Y(0, -2) \rightarrow (2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, -2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] matches [tex]\(Y(0, -2) \rightarrow (2, 0)\)[/tex]
- But [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0)\)[/tex] gets mapped to [tex]\((0, -2)\)[/tex]
- Therefore, the 90-degree rotation rule does not match both vertices accurately.
2. 180 Degree Rotation Counterclockwise ([tex]\(R_{0,180^\circ}\)[/tex]):
- For a 180-degree rotation counterclockwise about the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
- Applying this to the vertices:
[tex]\[ Y(0, -2) \rightarrow (0, 2) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (2, 0) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] should match, however, [tex]\(Y(0, -2) \rightarrow (0, 2)\)[/tex] does not match
- [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (2, 0)\)[/tex] does not match
- Therefore, the 180-degree rotation rule does not describe the transformation.
3. 270 Degree Rotation Counterclockwise ([tex]\(R_{0,270^\circ}\)[/tex]):
- For a 270-degree rotation counterclockwise about the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
- Applying this to the vertices:
[tex]\[ Y(0, -2) \rightarrow (-2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, 2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] does not match [tex]\(Y(0, -2) \rightarrow (-2, 0)\)[/tex]
- [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (0, 2)\)[/tex] does not match
- Therefore, the 270-degree rotation rule does not describe the transformation.
4. Rule [tex]\((x, y) \rightarrow (-y, x)\)[/tex]:
- This is the same transformation as a 90-degree rotation counterclockwise:
[tex]\[ Y(0, -2) \rightarrow (2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, -2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] matches [tex]\(Y(0, -2) \rightarrow (2, 0)\)[/tex]
- But [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (0, -2)\)[/tex]
- Therefore, this transformation does not describe the overall transformation.
5. Rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex]:
- This is the same transformation as a 270-degree rotation counterclockwise:
[tex]\[ Y(0, -2) \rightarrow (-2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, 2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] does not match [tex]\(Y(0, -2) \rightarrow (-2, 0)\)[/tex]
- [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (0, 2)\)[/tex]
- Therefore, this transformation does not describe the overall transformation.
Since none of the given transformations correctly map both [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex] to [tex]\(Y'\)[/tex] and [tex]\(Z'\)[/tex],
the transformations that map the original triangle accurately are not among the provided options.
Therefore, the transformations that could describe the rotation are none of the given options:
- [tex]\(R_{0,90^\circ}\)[/tex]
- [tex]\(R_{0,180^\circ}\)[/tex]
- [tex]\(R_{0,270^\circ}\)[/tex]
- [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
Hence, the answer would be an empty set.
1. 90 Degree Rotation Counterclockwise ([tex]\(R_{0,90^\circ}\)[/tex]):
- For a 90-degree rotation counterclockwise about the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- Applying this to the vertices:
[tex]\[ Y(0, -2) \rightarrow (2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, -2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] matches [tex]\(Y(0, -2) \rightarrow (2, 0)\)[/tex]
- But [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0)\)[/tex] gets mapped to [tex]\((0, -2)\)[/tex]
- Therefore, the 90-degree rotation rule does not match both vertices accurately.
2. 180 Degree Rotation Counterclockwise ([tex]\(R_{0,180^\circ}\)[/tex]):
- For a 180-degree rotation counterclockwise about the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
- Applying this to the vertices:
[tex]\[ Y(0, -2) \rightarrow (0, 2) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (2, 0) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] should match, however, [tex]\(Y(0, -2) \rightarrow (0, 2)\)[/tex] does not match
- [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (2, 0)\)[/tex] does not match
- Therefore, the 180-degree rotation rule does not describe the transformation.
3. 270 Degree Rotation Counterclockwise ([tex]\(R_{0,270^\circ}\)[/tex]):
- For a 270-degree rotation counterclockwise about the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
- Applying this to the vertices:
[tex]\[ Y(0, -2) \rightarrow (-2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, 2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] does not match [tex]\(Y(0, -2) \rightarrow (-2, 0)\)[/tex]
- [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (0, 2)\)[/tex] does not match
- Therefore, the 270-degree rotation rule does not describe the transformation.
4. Rule [tex]\((x, y) \rightarrow (-y, x)\)[/tex]:
- This is the same transformation as a 90-degree rotation counterclockwise:
[tex]\[ Y(0, -2) \rightarrow (2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, -2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] matches [tex]\(Y(0, -2) \rightarrow (2, 0)\)[/tex]
- But [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (0, -2)\)[/tex]
- Therefore, this transformation does not describe the overall transformation.
5. Rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex]:
- This is the same transformation as a 270-degree rotation counterclockwise:
[tex]\[ Y(0, -2) \rightarrow (-2, 0) \][/tex]
[tex]\[ Z(-2, 0) \rightarrow (0, 2) \][/tex]
- Comparing these with the image vertices:
- [tex]\(Y'(2, 0)\)[/tex] does not match [tex]\(Y(0, -2) \rightarrow (-2, 0)\)[/tex]
- [tex]\(Z'(2, -2)\)[/tex] should match, however, [tex]\(Z(-2, 0) \rightarrow (0, 2)\)[/tex]
- Therefore, this transformation does not describe the overall transformation.
Since none of the given transformations correctly map both [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex] to [tex]\(Y'\)[/tex] and [tex]\(Z'\)[/tex],
the transformations that map the original triangle accurately are not among the provided options.
Therefore, the transformations that could describe the rotation are none of the given options:
- [tex]\(R_{0,90^\circ}\)[/tex]
- [tex]\(R_{0,180^\circ}\)[/tex]
- [tex]\(R_{0,270^\circ}\)[/tex]
- [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
Hence, the answer would be an empty set.
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