Answer :
To determine which transformations could have taken place to move a vertex from [tex]\((3, -2)\)[/tex] to [tex]\((2, 3)\)[/tex], we need to consider the possible rotations around the origin and how these rotations affect the coordinates of the points.
1. Rotation [tex]\( R_{0, 90^{\circ}} \)[/tex]:
- A [tex]\(90^{\circ}\)[/tex] counterclockwise rotation transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-(-2), 3) = (2, 3) \][/tex]
- This matches the given rotated point [tex]\((2, 3)\)[/tex].
2. Rotation [tex]\( R_{0, 180^{\circ}} \)[/tex]:
- A [tex]\(180^{\circ}\)[/tex] counterclockwise rotation transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-3, 2) \][/tex]
- This does not match the given rotated point [tex]\((2, 3)\)[/tex].
3. Rotation [tex]\( R_{0, 270^{\circ}} \)[/tex]:
- A [tex]\(270^{\circ}\)[/tex] counterclockwise (or equivalently, [tex]\(90^{\circ}\)[/tex] clockwise) rotation transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-2, -3) \][/tex]
- This does not match the given rotated point [tex]\((2, 3)\)[/tex].
4. Rotation [tex]\( R_{0, -90^{\circ}} \)[/tex]:
- A [tex]\(90^{\circ}\)[/tex] clockwise rotation (equivalent to [tex]\(-90^{\circ}\)[/tex]) transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-2, -3) \][/tex]
- This does not match the given rotated point [tex]\((2, 3)\)[/tex].
5. Rotation [tex]\( R_{0, -270^{\circ}} \)[/tex]:
- A [tex]\(270^{\circ}\)[/tex] clockwise rotation (equivalent to [tex]\(-270^{\circ}\)[/tex]) transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex], which is the same as a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation.
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (2, 3) \][/tex]
- This matches the given rotated point [tex]\((2, 3)\)[/tex].
From these observations, we find that the following transformations could have taken place:
- [tex]\( R_{0, 90^{\circ}} \)[/tex]
- [tex]\( R_{0, -270^{\circ}} \)[/tex]
1. Rotation [tex]\( R_{0, 90^{\circ}} \)[/tex]:
- A [tex]\(90^{\circ}\)[/tex] counterclockwise rotation transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-(-2), 3) = (2, 3) \][/tex]
- This matches the given rotated point [tex]\((2, 3)\)[/tex].
2. Rotation [tex]\( R_{0, 180^{\circ}} \)[/tex]:
- A [tex]\(180^{\circ}\)[/tex] counterclockwise rotation transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-3, 2) \][/tex]
- This does not match the given rotated point [tex]\((2, 3)\)[/tex].
3. Rotation [tex]\( R_{0, 270^{\circ}} \)[/tex]:
- A [tex]\(270^{\circ}\)[/tex] counterclockwise (or equivalently, [tex]\(90^{\circ}\)[/tex] clockwise) rotation transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-2, -3) \][/tex]
- This does not match the given rotated point [tex]\((2, 3)\)[/tex].
4. Rotation [tex]\( R_{0, -90^{\circ}} \)[/tex]:
- A [tex]\(90^{\circ}\)[/tex] clockwise rotation (equivalent to [tex]\(-90^{\circ}\)[/tex]) transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (-2, -3) \][/tex]
- This does not match the given rotated point [tex]\((2, 3)\)[/tex].
5. Rotation [tex]\( R_{0, -270^{\circ}} \)[/tex]:
- A [tex]\(270^{\circ}\)[/tex] clockwise rotation (equivalent to [tex]\(-270^{\circ}\)[/tex]) transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex], which is the same as a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation.
- For the point [tex]\((3, -2)\)[/tex], applying this rotation would give:
[tex]\[ (2, 3) \][/tex]
- This matches the given rotated point [tex]\((2, 3)\)[/tex].
From these observations, we find that the following transformations could have taken place:
- [tex]\( R_{0, 90^{\circ}} \)[/tex]
- [tex]\( R_{0, -270^{\circ}} \)[/tex]
