To determine which rotation transformations could have lead to the final coordinates of the vertex, let's analyze the possible rotations around the origin (0,0):
1. Rotation by [tex]\(90^\circ\)[/tex] Counter-clockwise (CCW)
- The transformation for a [tex]\(90^\circ\)[/tex] CCW rotation is given by:
[tex]\[
(x, y) \rightarrow (-y, x)
\][/tex]
- Applying this transformation to [tex]\((3, -2)\)[/tex]:
[tex]\[
(3, -2) \rightarrow (-(-2), 3) = (2, 3)
\][/tex]
2. Rotation by [tex]\(180^\circ\)[/tex]
- The transformation for a [tex]\(180^\circ\)[/tex] rotation is given by:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
- Applying this transformation to [tex]\((3, -2)\)[/tex]:
[tex]\[
(3, -2) \rightarrow (-3, 2)
\][/tex]
- This does not match [tex]\((2, 3)\)[/tex].
3. Rotation by [tex]\(270^\circ\)[/tex] Counter-clockwise (CCW) (or equivalently [tex]\(-90^\circ\)[/tex] Clockwise (CW))
- The transformation for a [tex]\(270^\circ\)[/tex] CCW rotation is given by:
[tex]\[
(x, y) \rightarrow (y, -x)
\][/tex]
- Applying this transformation to [tex]\((3, -2)\)[/tex]:
[tex]\[
(3, -2) \rightarrow (-2, -3)
\][/tex]
- This does not match [tex]\((2, 3)\)[/tex].
4. Rotation by [tex]\(-90^\circ\)[/tex] Clockwise (CW) (or equivalently [tex]\(270^\circ\)[/tex] CCW)
- The transformation for a [tex]\(-90^\circ\)[/tex] CW rotation is given by:
[tex]\[
(x, y) \rightarrow (y, -x)
\][/tex]
- Applying this transformation to [tex]\((3, -2)\)[/tex]:
[tex]\[
(3, -2) \rightarrow (-2, -3)
\][/tex]
- This does not match [tex]\((2, 3)\)[/tex].
5. Rotation by [tex]\(-270^\circ\)[/tex] Clockwise (CW) (or equivalently [tex]\(90^\circ\)[/tex] CCW)
- The transformation for a [tex]\(-270^\circ\)[/tex] CW rotation is given by:
[tex]\[
(x, y) \rightarrow (-y, x)
\][/tex]
- Applying this transformation to [tex]\((3, -2)\)[/tex]:
[tex]\[
(3, -2) \rightarrow (2, 3)
\][/tex]
- This transformation matches [tex]\((2, 3)\)[/tex].
From these calculations, it is clear that the rotations that map the initial coordinates [tex]\((3, -2)\)[/tex] to the final coordinates [tex]\((2, 3)\)[/tex] are:
[tex]\[
R_{0,90^{\circ}} \quad \text{and} \quad R_{0,-270^{\circ}}
\][/tex]
So, the valid transformations are:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]
2. [tex]\(R_{0,-270^{\circ}}\)[/tex]
