To determine the transformations that could have taken place to move a vertex from [tex]\((0,5)\)[/tex] to [tex]\((5,0)\)[/tex], we need to consider the different rotations and their effects on coordinates.
1. Rotation [tex]\(R_{0,90^{\circ}}\)[/tex]: Rotating [tex]\((0,5)\)[/tex] by 90 degrees clockwise around the origin [tex]\((0,0)\)[/tex] will transform it into [tex]\((-5,0)\)[/tex].
2. Rotation [tex]\(R_{0,180^{\circ}}\)[/tex]: Rotating [tex]\((0,5)\)[/tex] by 180 degrees around the origin will transform it into [tex]\((0,-5)\)[/tex].
3. Rotation [tex]\(R_{0,270^{\circ}}\)[/tex]: Rotating [tex]\((0,5)\)[/tex] by 270 degrees clockwise (equivalent to -90 degrees counterclockwise) around the origin will transform it into [tex]\((5,0)\)[/tex].
4. Rotation [tex]\(R_{0,-90^{\circ}}\)[/tex]: Rotating [tex]\((0,5)\)[/tex] by -90 degrees counterclockwise (equivalent to 270 degrees clockwise) around the origin will also transform it into [tex]\((5,0)\)[/tex].
5. Rotation [tex]\(R_{0,-180^{\circ}}\)[/tex]: Rotating [tex]\((0,5)\)[/tex] by -180 degrees (which is the same as 180 degrees in either direction) around the origin will transform it into [tex]\((0,-5)\)[/tex].
From these transformations, the only ones that result in the vertex moving from [tex]\((0,5)\)[/tex] to [tex]\((5,0)\)[/tex] are:
- [tex]\(R_{0,270^{\circ}}\)[/tex]
- [tex]\(R_{0,-90^{\circ}}\)[/tex]
Therefore, the correct transformations that could have taken place are:
[tex]\[
\boxed{R_{0,270^{\circ}} \text{ and } R_{0,-90^{\circ}}}
\][/tex]
In terms of the options provided, these correspond to:
- [tex]\(R_{0,270^{\circ}}\)[/tex] is indicated by option 3.
- [tex]\(R_{0,-90^{\circ}}\)[/tex] is indicated by option 4.
So the selected options are 3 and 4.
