Answer :
Alright, let's analyze the problem step by step.
1. Identifying the original and rotated vertices:
- Original vertices:
- [tex]\( X(0,0) \)[/tex]
- [tex]\( Y(0,-2) \)[/tex]
- [tex]\( Z(-2,-2) \)[/tex]
- Rotated vertices:
- [tex]\( X^{\prime}(0,0) \)[/tex]
- [tex]\( Y^{\prime}(2,0) \)[/tex]
- [tex]\( Z^{\prime}(2,-2) \)[/tex]
2. Checking the 90° rotation rule [tex]\( R_{0,90^{\circ}} \)[/tex]:
- The general rule for a 90° rotation is: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (-0,0) = (0,0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (2,0) = Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2,-2) = Z' \)[/tex]
- All rotated points match [tex]\( (0,0), (2,0), (2,-2) \)[/tex].
- Hence, this rule fits the transformation.
3. Checking the 180° rotation rule [tex]\( R_{0,180^{\circ}} \)[/tex]:
- The general rule for a 180° rotation is: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (-0, -0) = (0,0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (0,2) \neq Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2,2) \neq Z' \)[/tex]
- Not all points match. This rule does not fit the transformation.
4. Checking the 270° rotation rule [tex]\( R_{0,270^{\circ}} \)[/tex]:
- The general rule for a 270° rotation is: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (0, 0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (-2, 0) \neq Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (-2,2) \neq Z' \)[/tex]
- Not all points match. This rule does not fit the transformation.
5. Checking the custom rule [tex]\( (x, y) \rightarrow (-y, x) \)[/tex]:
- This is the same as the 90° rotation rule already evaluated.
- As determined, all points match [tex]\( (0,0), (2,0), (2,-2) \)[/tex].
- Hence, this rule fits the transformation.
6. Checking the custom rule [tex]\( (x, y) \rightarrow (y, -x) \)[/tex]:
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (0, 0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (-2, 0) \neq Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (-2,2) \neq Z' \)[/tex]
- Not all points match. This rule does not fit the transformation.
Conclusion:
The rules that describe the rotation are:
- [tex]\( R_{0,90^{\circ}} \)[/tex]
- [tex]\( (x, y) \rightarrow(-y, x) \)[/tex]
These are the two options that correctly describe the rotation.
1. Identifying the original and rotated vertices:
- Original vertices:
- [tex]\( X(0,0) \)[/tex]
- [tex]\( Y(0,-2) \)[/tex]
- [tex]\( Z(-2,-2) \)[/tex]
- Rotated vertices:
- [tex]\( X^{\prime}(0,0) \)[/tex]
- [tex]\( Y^{\prime}(2,0) \)[/tex]
- [tex]\( Z^{\prime}(2,-2) \)[/tex]
2. Checking the 90° rotation rule [tex]\( R_{0,90^{\circ}} \)[/tex]:
- The general rule for a 90° rotation is: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (-0,0) = (0,0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (2,0) = Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2,-2) = Z' \)[/tex]
- All rotated points match [tex]\( (0,0), (2,0), (2,-2) \)[/tex].
- Hence, this rule fits the transformation.
3. Checking the 180° rotation rule [tex]\( R_{0,180^{\circ}} \)[/tex]:
- The general rule for a 180° rotation is: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (-0, -0) = (0,0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (0,2) \neq Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (2,2) \neq Z' \)[/tex]
- Not all points match. This rule does not fit the transformation.
4. Checking the 270° rotation rule [tex]\( R_{0,270^{\circ}} \)[/tex]:
- The general rule for a 270° rotation is: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (0, 0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (-2, 0) \neq Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (-2,2) \neq Z' \)[/tex]
- Not all points match. This rule does not fit the transformation.
5. Checking the custom rule [tex]\( (x, y) \rightarrow (-y, x) \)[/tex]:
- This is the same as the 90° rotation rule already evaluated.
- As determined, all points match [tex]\( (0,0), (2,0), (2,-2) \)[/tex].
- Hence, this rule fits the transformation.
6. Checking the custom rule [tex]\( (x, y) \rightarrow (y, -x) \)[/tex]:
- Applying this to the original vertices:
- [tex]\( X(0,0) \rightarrow (0, 0) = X' \)[/tex]
- [tex]\( Y(0,-2) \rightarrow (-2, 0) \neq Y' \)[/tex]
- [tex]\( Z(-2,-2) \rightarrow (-2,2) \neq Z' \)[/tex]
- Not all points match. This rule does not fit the transformation.
Conclusion:
The rules that describe the rotation are:
- [tex]\( R_{0,90^{\circ}} \)[/tex]
- [tex]\( (x, y) \rightarrow(-y, x) \)[/tex]
These are the two options that correctly describe the rotation.
