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To determine which rules describe the rotation of triangle [tex]\( XYZ \)[/tex] to its image [tex]\( X'Y'Z' \)[/tex], we need to examine the coordinates of the original and rotated triangles. Here’s a step-by-step analysis:
1. Original Triangle [tex]\( XYZ \)[/tex] Coordinates:
- [tex]\( X(0,0) \)[/tex]
- [tex]\( Y(0,-2) \)[/tex]
- [tex]\( Z(-2,-2) \)[/tex]
2. Image Triangle [tex]\( X'Y'Z' \)[/tex] Coordinates:
- [tex]\( X'(0,0) \)[/tex]
- [tex]\( Y'(2,0) \)[/tex]
- [tex]\( Z'(2,-2) \)[/tex]
3. Examine Possible Rotations:
- Rotation by [tex]\( 90^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,90^\circ} \)[/tex]):
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this to the original coordinates:
- [tex]\( X(0,0) \rightarrow (0, 0) \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (2, 0) \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2, 2) \)[/tex]
However, [tex]\( Z' = (2, -2) \)[/tex], not [tex]\( (2, 2) \)[/tex]. So, rotation by [tex]\( 90^\circ \)[/tex] does not match.
- Rotation by [tex]\( 180^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,180^\circ} \)[/tex]):
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Applying this to the original coordinates:
- [tex]\( X(0,0) \rightarrow (0,0) \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (0,2) \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2,2) \)[/tex]
However, [tex]\( Y' = (2, 0) \)[/tex] and [tex]\( Z' = (2, -2) \)[/tex]. So, rotation by [tex]\( 180^\circ \)[/tex] does not match.
- Rotation by [tex]\( 270^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,270^\circ} \)[/tex]):
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this to the original coordinates:
- [tex]\( X(0,0) \rightarrow (0, 0) \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (-2, 0) \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (-2, 2) \)[/tex]
However, [tex]\( Y' = (2, 0) \)[/tex] and [tex]\( Z' = (2, -2) \)[/tex]. So, rotation by [tex]\( 270^\circ \)[/tex] does not match.
Thus, none of the given rotations [tex]\( R_{0,90^\circ} \)[/tex], [tex]\( R_{0,180^\circ} \)[/tex], or [tex]\( R_{0,270^\circ} \)[/tex] correctly describe the rotation that maps [tex]\( XYZ \)[/tex] to [tex]\( X'Y'Z' \)[/tex].
4. Examining Other Rules:
- Rule: [tex]\( (x, y) \rightarrow (-y, x) \)[/tex]:
This corresponds to a [tex]\( 90^\circ \)[/tex] counterclockwise rotation mentioned above and has already been shown not to match.
- Rule: [tex]\( (x, y) \rightarrow (y,-x) \)[/tex]:
This corresponds to a [tex]\( 270^\circ \)[/tex] counterclockwise rotation mentioned above and has already been shown not to match.
Therefore, the answer is that there are no rotations among the given options that correctly describe the transformation of the triangle. The correct set of rules and rotations does not fulfill the transformation requirements given the problem’s constraints.
Given the detailed analysis, there are no valid options among the listed rotations or other rules that describe the rotation. Therefore, there are no correct rotation rules provided for this specific transformation.
1. Original Triangle [tex]\( XYZ \)[/tex] Coordinates:
- [tex]\( X(0,0) \)[/tex]
- [tex]\( Y(0,-2) \)[/tex]
- [tex]\( Z(-2,-2) \)[/tex]
2. Image Triangle [tex]\( X'Y'Z' \)[/tex] Coordinates:
- [tex]\( X'(0,0) \)[/tex]
- [tex]\( Y'(2,0) \)[/tex]
- [tex]\( Z'(2,-2) \)[/tex]
3. Examine Possible Rotations:
- Rotation by [tex]\( 90^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,90^\circ} \)[/tex]):
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this to the original coordinates:
- [tex]\( X(0,0) \rightarrow (0, 0) \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (2, 0) \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2, 2) \)[/tex]
However, [tex]\( Z' = (2, -2) \)[/tex], not [tex]\( (2, 2) \)[/tex]. So, rotation by [tex]\( 90^\circ \)[/tex] does not match.
- Rotation by [tex]\( 180^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,180^\circ} \)[/tex]):
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Applying this to the original coordinates:
- [tex]\( X(0,0) \rightarrow (0,0) \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (0,2) \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2,2) \)[/tex]
However, [tex]\( Y' = (2, 0) \)[/tex] and [tex]\( Z' = (2, -2) \)[/tex]. So, rotation by [tex]\( 180^\circ \)[/tex] does not match.
- Rotation by [tex]\( 270^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,270^\circ} \)[/tex]):
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this to the original coordinates:
- [tex]\( X(0,0) \rightarrow (0, 0) \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (-2, 0) \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (-2, 2) \)[/tex]
However, [tex]\( Y' = (2, 0) \)[/tex] and [tex]\( Z' = (2, -2) \)[/tex]. So, rotation by [tex]\( 270^\circ \)[/tex] does not match.
Thus, none of the given rotations [tex]\( R_{0,90^\circ} \)[/tex], [tex]\( R_{0,180^\circ} \)[/tex], or [tex]\( R_{0,270^\circ} \)[/tex] correctly describe the rotation that maps [tex]\( XYZ \)[/tex] to [tex]\( X'Y'Z' \)[/tex].
4. Examining Other Rules:
- Rule: [tex]\( (x, y) \rightarrow (-y, x) \)[/tex]:
This corresponds to a [tex]\( 90^\circ \)[/tex] counterclockwise rotation mentioned above and has already been shown not to match.
- Rule: [tex]\( (x, y) \rightarrow (y,-x) \)[/tex]:
This corresponds to a [tex]\( 270^\circ \)[/tex] counterclockwise rotation mentioned above and has already been shown not to match.
Therefore, the answer is that there are no rotations among the given options that correctly describe the transformation of the triangle. The correct set of rules and rotations does not fulfill the transformation requirements given the problem’s constraints.
Given the detailed analysis, there are no valid options among the listed rotations or other rules that describe the rotation. Therefore, there are no correct rotation rules provided for this specific transformation.
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