Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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]
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]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
