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Given a vertex of a polygon located initially at [tex]\((3, -2)\)[/tex], we observe that after a rotation, the vertex is located at [tex]\((2, 3)\)[/tex]. We need to determine which transformations could have taken place. We will examine each of the given potential rotations.
1. Rotation [tex]\(R_{0, 90^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(90^{\circ}\)[/tex] counter-clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-(-2), 3) \rightarrow (2, 3) \][/tex]
Thus, one valid transformation is [tex]\(R_{0, 90^{\circ}}\)[/tex].
2. Rotation [tex]\(R_{0, 180^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(180^{\circ}\)[/tex] counter-clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-3, 2) \][/tex]
This doesn't match our target point [tex]\((2, 3)\)[/tex]. So this transformation is not valid.
3. Rotation [tex]\(R_{0, 270^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(270^{\circ}\)[/tex] counter-clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
This doesn't match our target point [tex]\((2, 3)\)[/tex]. So this transformation is not valid.
4. Rotation [tex]\(R_{0, -90^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(-90^{\circ}\)[/tex] counter-clockwise (which is the same as rotating [tex]\(270^{\circ}\)[/tex] clockwise) around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
This doesn't match our target point [tex]\((2, 3)\)[/tex]. So this transformation is not valid.
5. Rotation [tex]\(R_{0, -270^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(-270^{\circ}\)[/tex] counter-clockwise (which is the same as rotating [tex]\(90^{\circ}\)[/tex] clockwise) around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-(-2), 3) \rightarrow (2, 3) \][/tex]
Thus, another valid transformation is [tex]\(R_{0, -270^{\circ}}\)[/tex].
Therefore, the two possible transformations that could have moved the vertex 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 correct options are:
- [tex]\(R_{0, 90^{\circ}}\)[/tex]
- [tex]\(R_{0, -270^{\circ}}\)[/tex]
1. Rotation [tex]\(R_{0, 90^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(90^{\circ}\)[/tex] counter-clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-(-2), 3) \rightarrow (2, 3) \][/tex]
Thus, one valid transformation is [tex]\(R_{0, 90^{\circ}}\)[/tex].
2. Rotation [tex]\(R_{0, 180^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(180^{\circ}\)[/tex] counter-clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-3, 2) \][/tex]
This doesn't match our target point [tex]\((2, 3)\)[/tex]. So this transformation is not valid.
3. Rotation [tex]\(R_{0, 270^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(270^{\circ}\)[/tex] counter-clockwise around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
This doesn't match our target point [tex]\((2, 3)\)[/tex]. So this transformation is not valid.
4. Rotation [tex]\(R_{0, -90^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(-90^{\circ}\)[/tex] counter-clockwise (which is the same as rotating [tex]\(270^{\circ}\)[/tex] clockwise) around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
This doesn't match our target point [tex]\((2, 3)\)[/tex]. So this transformation is not valid.
5. Rotation [tex]\(R_{0, -270^{\circ}}\)[/tex]:
The formula to rotate a point [tex]\((x, y)\)[/tex] by [tex]\(-270^{\circ}\)[/tex] counter-clockwise (which is the same as rotating [tex]\(90^{\circ}\)[/tex] clockwise) around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Applying this to our initial point [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-(-2), 3) \rightarrow (2, 3) \][/tex]
Thus, another valid transformation is [tex]\(R_{0, -270^{\circ}}\)[/tex].
Therefore, the two possible transformations that could have moved the vertex 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 correct options are:
- [tex]\(R_{0, 90^{\circ}}\)[/tex]
- [tex]\(R_{0, -270^{\circ}}\)[/tex]
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