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To determine which rotation rules could describe the transformation of triangle [tex]\( XYZ \)[/tex] with vertices [tex]\( X(0, 0) \)[/tex], [tex]\( Y(0, -2) \)[/tex], and [tex]\( Z(-2, 2) \)[/tex] to the image triangle [tex]\( X'(0, 0) \)[/tex], [tex]\( Y'(2, 0) \)[/tex], and [tex]\( Z'(2, -2) \)[/tex], we need to examine the possible rotations and their effects on the vertices.
1. Initial and Image Vertices:
- Original vertices: [tex]\( X(0, 0) \)[/tex], [tex]\( Y(0, -2) \)[/tex], and [tex]\( Z(-2, 2) \)[/tex].
- Rotated vertices: [tex]\( X'(0, 0) \)[/tex], [tex]\( Y'(2, 0) \)[/tex], and [tex]\( Z'(2, -2) \)[/tex].
2. Understanding Rotation by 90 Degrees Counterclockwise:
- Rotation by 90 degrees counterclockwise transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- Applying this to [tex]\( Y(0, -2) \)[/tex]: [tex]\( (0, -2) \rightarrow (2, 0) \)[/tex]. This matches [tex]\( Y'(2, 0) \)[/tex].
- Applying this to [tex]\( Z(-2, 2) \)[/tex]: [tex]\( (-2, 2) \rightarrow (-2, -2) \)[/tex]. This does not match [tex]\( Z'(2, -2) \)[/tex].
3. Understanding Rotation by 270 Degrees Counterclockwise (or 90 Degrees Clockwise):
- Rotation by 270 degrees counterclockwise transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
- Applying this to [tex]\( Y(0, -2) \)[/tex]: [tex]\( (0, -2) \rightarrow (-2, 0) \)[/tex]. This does not match [tex]\( Y'(2, 0) \)[/tex].
- Applying this to [tex]\( Z(-2, 2) \)[/tex]: [tex]\( (-2, 2) \rightarrow (2, 2) \)[/tex]. This does not match [tex]\( Z'(2, -2) \)[/tex].
4. Understanding Rotation by 180 Degrees:
- Rotation by 180 degrees transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- Applying this to [tex]\( Y(0, -2) \)[/tex]: [tex]\( (0, -2) \rightarrow (0, 2) \)[/tex]. This does not match [tex]\( Y'(2, 0) \)[/tex].
- Applying this to [tex]\( Z(-2, 2) \)[/tex]: [tex]\( (-2, 2) \rightarrow (2, -2) \)[/tex]. This matches [tex]\( Z'(2, -2) \)[/tex], but the mismatch with [tex]\( Y'(2, 0) \)[/tex] discards this.
5. Verification of Other Rotation Rules:
- [tex]\((x, y) \rightarrow (-y, x)\)[/tex]: This corresponds to a 90 degrees counterclockwise rotation.
- [tex]\((x, y) \rightarrow (y, -x)\)[/tex]: This corresponds to a 270 degrees counterclockwise (or 90 degrees clockwise) rotation.
Given the above examination, neither rotation by 90 degrees, 180 degrees, nor 270 degrees provides a complete match for all vertices because both rotations only partially align with the transformed vertices.
Thus, after verifying all possible rotations, we conclude that none of the given options fully describe the observed transformation.
Therefore, there are no valid rotation rules that exactly match the transformation of triangle [tex]\( XYZ \)[/tex] to triangle [tex]\( X'Y'Z' \)[/tex]. The result shows that:
[tex]\[ \boxed{[]} \][/tex]
The rotation providing accurate results does not exist in the given options.
1. Initial and Image Vertices:
- Original vertices: [tex]\( X(0, 0) \)[/tex], [tex]\( Y(0, -2) \)[/tex], and [tex]\( Z(-2, 2) \)[/tex].
- Rotated vertices: [tex]\( X'(0, 0) \)[/tex], [tex]\( Y'(2, 0) \)[/tex], and [tex]\( Z'(2, -2) \)[/tex].
2. Understanding Rotation by 90 Degrees Counterclockwise:
- Rotation by 90 degrees counterclockwise transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- Applying this to [tex]\( Y(0, -2) \)[/tex]: [tex]\( (0, -2) \rightarrow (2, 0) \)[/tex]. This matches [tex]\( Y'(2, 0) \)[/tex].
- Applying this to [tex]\( Z(-2, 2) \)[/tex]: [tex]\( (-2, 2) \rightarrow (-2, -2) \)[/tex]. This does not match [tex]\( Z'(2, -2) \)[/tex].
3. Understanding Rotation by 270 Degrees Counterclockwise (or 90 Degrees Clockwise):
- Rotation by 270 degrees counterclockwise transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
- Applying this to [tex]\( Y(0, -2) \)[/tex]: [tex]\( (0, -2) \rightarrow (-2, 0) \)[/tex]. This does not match [tex]\( Y'(2, 0) \)[/tex].
- Applying this to [tex]\( Z(-2, 2) \)[/tex]: [tex]\( (-2, 2) \rightarrow (2, 2) \)[/tex]. This does not match [tex]\( Z'(2, -2) \)[/tex].
4. Understanding Rotation by 180 Degrees:
- Rotation by 180 degrees transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- Applying this to [tex]\( Y(0, -2) \)[/tex]: [tex]\( (0, -2) \rightarrow (0, 2) \)[/tex]. This does not match [tex]\( Y'(2, 0) \)[/tex].
- Applying this to [tex]\( Z(-2, 2) \)[/tex]: [tex]\( (-2, 2) \rightarrow (2, -2) \)[/tex]. This matches [tex]\( Z'(2, -2) \)[/tex], but the mismatch with [tex]\( Y'(2, 0) \)[/tex] discards this.
5. Verification of Other Rotation Rules:
- [tex]\((x, y) \rightarrow (-y, x)\)[/tex]: This corresponds to a 90 degrees counterclockwise rotation.
- [tex]\((x, y) \rightarrow (y, -x)\)[/tex]: This corresponds to a 270 degrees counterclockwise (or 90 degrees clockwise) rotation.
Given the above examination, neither rotation by 90 degrees, 180 degrees, nor 270 degrees provides a complete match for all vertices because both rotations only partially align with the transformed vertices.
Thus, after verifying all possible rotations, we conclude that none of the given options fully describe the observed transformation.
Therefore, there are no valid rotation rules that exactly match the transformation of triangle [tex]\( XYZ \)[/tex] to triangle [tex]\( X'Y'Z' \)[/tex]. The result shows that:
[tex]\[ \boxed{[]} \][/tex]
The rotation providing accurate results does not exist in the given options.
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