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Sagot :
Certainly! Let's determine which transformations could result in the vertex of the triangle moving from [tex]\((0,5)\)[/tex] to [tex]\((5,0)\)[/tex].
### Transformation Descriptions:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: A rotation by [tex]\(90^\circ\)[/tex] counterclockwise around the origin.
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: A rotation by [tex]\(180^\circ\)[/tex] around the origin.
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: A rotation by [tex]\(270^\circ\)[/tex] counterclockwise around the origin.
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: A rotation by [tex]\(90^\circ\)[/tex] clockwise around the origin.
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: A rotation by [tex]\(180^\circ\)[/tex] clockwise around the origin.
### Applying Transformations:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: Rotation by [tex]\(90^\circ\)[/tex] counterclockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((-5,0)\)[/tex]
- Result: [tex]\((-5,0)\)[/tex], which is not [tex]\((5,0)\)[/tex].
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: Rotation by [tex]\(180^\circ\)[/tex]
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((0,-5)\)[/tex]
- Result: [tex]\((0,-5)\)[/tex], which is not [tex]\((5,0)\)[/tex].
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: Rotation by [tex]\(270^\circ\)[/tex] counterclockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((5,0)\)[/tex]
- Result: [tex]\((5,0)\)[/tex], which matches the transformed coordinates.
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: Rotation by [tex]\(90^\circ\)[/tex] clockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((5,0)\)[/tex]
- Result: [tex]\((5,0)\)[/tex], which matches the transformed coordinates.
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: Rotation by [tex]\(180^\circ\)[/tex] clockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((0,-5)\)[/tex]
- Result: [tex]\((0,-5)\)[/tex], which is not [tex]\((5,0)\)[/tex].
### Conclusion:
The valid transformations are [tex]\(R_{0,270^{\circ}}\)[/tex] and [tex]\(R_{0,-90^{\circ}}\)[/tex], as they both result in the vertex moving from [tex]\((0,5)\)[/tex] to [tex]\((5,0)\)[/tex].
Hence, the correct options are:
- [tex]\(R_{0,270^{\circ}}\)[/tex]
- [tex]\(R_{0,-90^{\circ}}\)[/tex]
### Transformation Descriptions:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: A rotation by [tex]\(90^\circ\)[/tex] counterclockwise around the origin.
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: A rotation by [tex]\(180^\circ\)[/tex] around the origin.
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: A rotation by [tex]\(270^\circ\)[/tex] counterclockwise around the origin.
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: A rotation by [tex]\(90^\circ\)[/tex] clockwise around the origin.
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: A rotation by [tex]\(180^\circ\)[/tex] clockwise around the origin.
### Applying Transformations:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: Rotation by [tex]\(90^\circ\)[/tex] counterclockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((-5,0)\)[/tex]
- Result: [tex]\((-5,0)\)[/tex], which is not [tex]\((5,0)\)[/tex].
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: Rotation by [tex]\(180^\circ\)[/tex]
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((0,-5)\)[/tex]
- Result: [tex]\((0,-5)\)[/tex], which is not [tex]\((5,0)\)[/tex].
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: Rotation by [tex]\(270^\circ\)[/tex] counterclockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((5,0)\)[/tex]
- Result: [tex]\((5,0)\)[/tex], which matches the transformed coordinates.
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: Rotation by [tex]\(90^\circ\)[/tex] clockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((5,0)\)[/tex]
- Result: [tex]\((5,0)\)[/tex], which matches the transformed coordinates.
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: Rotation by [tex]\(180^\circ\)[/tex] clockwise
- Starting point: [tex]\((0,5)\)[/tex]
- Transformation to: [tex]\((0,-5)\)[/tex]
- Result: [tex]\((0,-5)\)[/tex], which is not [tex]\((5,0)\)[/tex].
### Conclusion:
The valid transformations are [tex]\(R_{0,270^{\circ}}\)[/tex] and [tex]\(R_{0,-90^{\circ}}\)[/tex], as they both result in the vertex moving from [tex]\((0,5)\)[/tex] to [tex]\((5,0)\)[/tex].
Hence, the correct options are:
- [tex]\(R_{0,270^{\circ}}\)[/tex]
- [tex]\(R_{0,-90^{\circ}}\)[/tex]
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