Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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]
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
