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Sagot :
To determine which transformations could have taken place, let's consider the different rotations around the origin and their effects on the coordinates of a point.
We initially have a vertex at [tex]\((3, -2)\)[/tex] and, after rotation, it is located at [tex]\((2, 3)\)[/tex].
### Step-by-Step Transformation Checks
1. Rotation by [tex]\(90^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
- This matches the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,90^{\circ}}\)[/tex] is a possibility.
2. Rotation by [tex]\(180^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-3, 2) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,180^{\circ}}\)[/tex] is not a valid solution.
3. Rotation by [tex]\(270^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,270^{\circ}}\)[/tex] is not a valid solution.
4. Rotation by [tex]\(-90^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,-90^{\circ}}\)[/tex] is not a valid solution.
5. Rotation by [tex]\(-270^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
- This matches the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,-270^{\circ}}\)[/tex] is a possibility.
### Conclusion
The two rotations that result in the given transformation from [tex]\((3, -2)\)[/tex] to [tex]\((2, 3)\)[/tex] are:
- [tex]\(R_{0,90^{\circ}}\)[/tex]
- [tex]\(R_{0,-270^{\circ}}\)[/tex]
Thus, the selected options are:
[tex]\[R_{0,90^{\circ}}\][/tex]
[tex]\[R_{0,-270^{\circ}}\][/tex]
We initially have a vertex at [tex]\((3, -2)\)[/tex] and, after rotation, it is located at [tex]\((2, 3)\)[/tex].
### Step-by-Step Transformation Checks
1. Rotation by [tex]\(90^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
- This matches the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,90^{\circ}}\)[/tex] is a possibility.
2. Rotation by [tex]\(180^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-3, 2) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,180^{\circ}}\)[/tex] is not a valid solution.
3. Rotation by [tex]\(270^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,270^{\circ}}\)[/tex] is not a valid solution.
4. Rotation by [tex]\(-90^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,-90^{\circ}}\)[/tex] is not a valid solution.
5. Rotation by [tex]\(-270^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
- This matches the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,-270^{\circ}}\)[/tex] is a possibility.
### Conclusion
The two rotations that result in the given transformation from [tex]\((3, -2)\)[/tex] to [tex]\((2, 3)\)[/tex] are:
- [tex]\(R_{0,90^{\circ}}\)[/tex]
- [tex]\(R_{0,-270^{\circ}}\)[/tex]
Thus, the selected options are:
[tex]\[R_{0,90^{\circ}}\][/tex]
[tex]\[R_{0,-270^{\circ}}\][/tex]
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