To determine which transformations can move a point [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex], we need to understand the nature of the transformations involved. Specifically, we will focus on rotations around the origin ([tex]\(0, 0\)[/tex]).
1. [tex]$\mathbf{R_{0,90^{\circ}}}$[/tex]:
- A 90-degree clockwise rotation about the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
- Applying this to the point [tex]\((0, 5)\)[/tex], we get:
[tex]\[
(0, 5) \rightarrow (5, 0)
\][/tex]
- Hence, a 90-degree clockwise rotation could indeed move [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex].
2. [tex]$\mathbf{R_{0,180^{\circ}}}$[/tex]:
- A 180-degree rotation about the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- Applying this to the point [tex]\((0, 5)\)[/tex], we get:
[tex]\[
(0, 5) \rightarrow (0, -5)
\][/tex]
- Therefore, a 180-degree rotation does not move [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex].
3. [tex]$\mathbf{R_{0,270^{\circ}}}$[/tex]:
- A 270-degree clockwise (or equivalently, 90-degree counterclockwise) rotation about the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- Applying this to the point [tex]\((0, 5)\)[/tex], we get:
[tex]\[
(0, 5) \rightarrow (5, 0)
\][/tex]
- Hence, a 270-degree clockwise rotation could indeed move [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex].
4. [tex]$\mathbf{R_{0,-90^{\circ}}}$[/tex]:
- A -90-degree (or 270-degree counterclockwise) rotation about the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- Applying this to the point [tex]\((0, 5)\)[/tex], we get:
[tex]\[
(0, 5) \rightarrow (5, 0)
\][/tex]
- Hence, a -90-degree rotation could indeed move [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex].
5. [tex]$\mathbf{R_{0,-180^{\circ}}}$[/tex]:
- A -180-degree (or 180-degree clockwise) rotation about the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- Applying this to the point [tex]\((0, 5)\)[/tex], we get:
[tex]\[
(0, 5) \rightarrow (0, -5)
\][/tex]
- Therefore, a -180-degree rotation does not move [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex].
Considering these explanations, the transformations that can move the vertex from [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex] are:
- [tex]$R_{0,90^{\circ}}$[/tex]
- [tex]$R_{0,270^{\circ}}$[/tex]
Thus, the correct options are:
[tex]\[
\boxed{R_{0,90^{\circ}} \text{ and } R_{0,270^{\circ}}}
\][/tex]
