Answered

Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

One vertex of a polygon is located at [tex]\((3, -2)\)[/tex]. After a rotation, the vertex is located at [tex]\((2, 3)\)[/tex].

Which transformations could have taken place? Select two options.

A. [tex]\(R_{0, 90^{\circ}}\)[/tex]
B. [tex]\(R_{0, 180^{\circ}}\)[/tex]
C. [tex]\(R_{0, 270^{\circ}}\)[/tex]
D. [tex]\(R_{0, -90^{\circ}}\)[/tex]
E. [tex]\(R_{0, -270^{\circ}}\)[/tex]

Sagot :

GinnyAnswer
To determine which transformations could have taken place, let's consider the different rotations around the origin and their effects on the coordinates of a point.

We initially have a vertex at [tex]\((3, -2)\)[/tex] and, after rotation, it is located at [tex]\((2, 3)\)[/tex].

### Step-by-Step Transformation Checks

1. Rotation by [tex]\(90^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
- This matches the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,90^{\circ}}\)[/tex] is a possibility.

2. Rotation by [tex]\(180^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-3, 2) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,180^{\circ}}\)[/tex] is not a valid solution.

3. Rotation by [tex]\(270^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,270^{\circ}}\)[/tex] is not a valid solution.

4. Rotation by [tex]\(-90^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (-2, -3) \][/tex]
- This does not match the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,-90^{\circ}}\)[/tex] is not a valid solution.

5. Rotation by [tex]\(-270^\circ\)[/tex]:
- Formula: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[ (3, -2) \rightarrow (2, 3) \][/tex]
- This matches the rotated vertex [tex]\((2, 3)\)[/tex]. Therefore, [tex]\(R_{0,-270^{\circ}}\)[/tex] is a possibility.

### Conclusion
The two rotations that result in the given transformation 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 selected options are:
[tex]\[R_{0,90^{\circ}}\][/tex]
[tex]\[R_{0,-270^{\circ}}\][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.