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The 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 :

GinnyAnsweridn
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]
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