To determine which rotations cause the vertex [tex]\((3, -2)\)[/tex] to become [tex]\((2, 3)\)[/tex], we need to examine each possible rotation and see if it results in the given transformation.
1. Rotation [tex]\(R_{0,90^{\circ}}\)[/tex]:
- A [tex]\(90^{\circ}\)[/tex] counterclockwise rotation transforms a point [tex]\((x, y)\)[/tex] into [tex]\((-y, x)\)[/tex].
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[
(-(-2), 3) = (2, 3)
\][/tex]
- This matches the target coordinates [tex]\((2, 3)\)[/tex]. So, [tex]\(R_{0,90^{\circ}}\)[/tex] is a correct option.
2. Rotation [tex]\(R_{0,180^{\circ}}\)[/tex]:
- A [tex]\(180^{\circ}\)[/tex] counterclockwise rotation transforms a point [tex]\((x, y)\)[/tex] into [tex]\((-x, -y)\)[/tex].
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[
(-3, 2) \neq (2, 3)
\][/tex]
- This does not match the target coordinates. So, [tex]\(R_{0,180^{\circ}}\)[/tex] is not a correct option.
3. Rotation [tex]\(R_{0,270^{\circ}}\)[/tex]:
- A [tex]\(270^{\circ}\)[/tex] counterclockwise rotation transforms a point [tex]\((x, y)\)[/tex] into [tex]\((y, -x)\)[/tex].
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[
(-2, -3) \neq (2, 3)
\][/tex]
- This does not match the target coordinates. So, [tex]\(R_{0,270^{\circ}}\)[/tex] is not a correct option.
4. Rotation [tex]\(R_{0,-90^{\circ}}\)[/tex]:
- A [tex]\(-90^{\circ}\)[/tex] (or 270 degrees clockwise) rotation transforms a point [tex]\((x, y)\)[/tex] into [tex]\((y, -x)\)[/tex].
- Note that this is the same transformation as [tex]\(R_{0,270^{\circ}}\)[/tex].
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[
(-2, -3) \neq (2, 3)
\][/tex]
- This does not match the target coordinates. So, [tex]\(R_{0,-90^{\circ}}\)[/tex] is not a correct option.
5. Rotation [tex]\(R_{0,-270^{\circ}}\)[/tex]:
- A [tex]\(-270^{\circ}\)[/tex] (or 90 degrees clockwise) rotation transforms a point [tex]\((x, y)\)[/tex] into [tex]\((-y, x)\)[/tex].
- Note that this is the same transformation as [tex]\(R_{0,90^{\circ}}\)[/tex].
- Applying it to [tex]\((3, -2)\)[/tex]:
[tex]\[
(-(-2), 3) = (2, 3)
\][/tex]
- This matches the target coordinates [tex]\((2, 3)\)[/tex]. So, [tex]\(R_{0,-270^{\circ}}\)[/tex] is a correct option.
Therefore, the transformations that cause the vertex [tex]\((3, -2)\)[/tex] to become [tex]\((2, 3)\)[/tex] are:
- [tex]\(R_{0,90^{\circ}}\)[/tex]
- [tex]\(R_{0,-270^{\circ}}\)[/tex]
