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Sagot :
To answer this question, we need to determine which rotation transformations could have moved the point [tex]\((3, -2)\)[/tex] to the point [tex]\((2, 3)\)[/tex]. We will analyze the effects of rotating the point by 90°, 180°, 270°, -90°, and -270°.
### Rotation by 90° Clockwise ([tex]\(R_{0,90^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 90° clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-(-2), 3) \][/tex]
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
Thus, rotation by [tex]\(90^\circ\)[/tex] clockwise does result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 180° ([tex]\(R_{0,180^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 180° around the origin is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-3, 2) \][/tex]
Thus, rotation by [tex]\(180^\circ\)[/tex] does not result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 270° Clockwise ([tex]\(R_{0,270^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 270° clockwise (or equivalently 90° counterclockwise) around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
Thus, rotation by [tex]\(270^\circ\)[/tex] clockwise does not result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 90° Counterclockwise ([tex]\(R_{0,-90^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 90° counterclockwise around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
Thus, rotation by [tex]\(90^\circ\)[/tex] counterclockwise does not result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 270° Counterclockwise ([tex]\(R_{0,-270^{\circ}}\)[/tex])
Rotation by [tex]\(270^\circ\)[/tex] counterclockwise is the same as rotating by [tex]\(90^\circ\)[/tex] clockwise:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
Thus, rotation by [tex]\(270^\circ\)[/tex] counterclockwise results in the point [tex]\((2, 3)\)[/tex].
Therefore, the transformations that could have taken place are:
1. [tex]\(R_{0,90^\circ}\)[/tex]
2. [tex]\(R_{0,-270^\circ}\)[/tex]
Thus, the correct answer is:
[tex]\[R_{0,90^{\circ}}\][/tex]
[tex]\[R_{0,-270^{\circ}}\][/tex]
### Rotation by 90° Clockwise ([tex]\(R_{0,90^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 90° clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-(-2), 3) \][/tex]
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
Thus, rotation by [tex]\(90^\circ\)[/tex] clockwise does result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 180° ([tex]\(R_{0,180^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 180° around the origin is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-3, 2) \][/tex]
Thus, rotation by [tex]\(180^\circ\)[/tex] does not result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 270° Clockwise ([tex]\(R_{0,270^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 270° clockwise (or equivalently 90° counterclockwise) around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
Thus, rotation by [tex]\(270^\circ\)[/tex] clockwise does not result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 90° Counterclockwise ([tex]\(R_{0,-90^{\circ}}\)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by 90° counterclockwise around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
Thus, rotation by [tex]\(90^\circ\)[/tex] counterclockwise does not result in the point [tex]\((2, 3)\)[/tex].
### Rotation by 270° Counterclockwise ([tex]\(R_{0,-270^{\circ}}\)[/tex])
Rotation by [tex]\(270^\circ\)[/tex] counterclockwise is the same as rotating by [tex]\(90^\circ\)[/tex] clockwise:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this transformation to the point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
Thus, rotation by [tex]\(270^\circ\)[/tex] counterclockwise results in the point [tex]\((2, 3)\)[/tex].
Therefore, the transformations that could have taken place are:
1. [tex]\(R_{0,90^\circ}\)[/tex]
2. [tex]\(R_{0,-270^\circ}\)[/tex]
Thus, the correct answer is:
[tex]\[R_{0,90^{\circ}}\][/tex]
[tex]\[R_{0,-270^{\circ}}\][/tex]
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