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Given the vertices of triangle [tex]\(XYZ\)[/tex] as [tex]\(X(0,0)\)[/tex], [tex]\(Y(0,-2)\)[/tex], and [tex]\(Z(-2,2)\)[/tex], and the vertices of the image triangle [tex]\(X^{\prime}(0,0)\)[/tex], [tex]\(Y^{\prime}(2,0)\)[/tex], and [tex]\(Z^{\prime}(2,-2)\)[/tex], we need to determine which rotations might transform [tex]\(XYZ\)[/tex] to [tex]\(X^{\prime}Y^{\prime}Z^{\prime}\)[/tex].
We are provided several options:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]
2. [tex]\(R_{0,180^{\circ}}\)[/tex]
3. [tex]\(R_{0,270^{\circ}}\)[/tex]
4. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
5. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
Step-by-Step Analysis:
1. Test [tex]\(R_{0, 90^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (-y, x)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (-0, 0) = (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (2, 0)\)[/tex] which matches [tex]\(Y^{\prime}(2, 0)\)[/tex].
- [tex]\(Z(-2, 2) \rightarrow (-2, -2)\)[/tex], but [tex]\(Z^{\prime} is (2, -2)\)[/tex], so this transformation does not match.
2. Test [tex]\(R_{0, 180^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (0, 2)\)[/tex] which does not match [tex]\(Y^{\prime}(2, 0)\)[/tex].
- Therefore, this transformation does not match.
3. Test [tex]\(R_{0, 270^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (y, -x)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (-2, 0)\)[/tex] which does not match [tex]\(Y^{\prime}(2, 0)\)[/tex].
- Therefore, this transformation does not match.
Having checked all possible rotations and their coordinate transformation rules, we see that no combination of the listed rotations can transform triangle [tex]\(XYZ\)[/tex] to triangle [tex]\(X^{\prime}Y^{\prime}Z^{\prime}\)[/tex] correctly.
Thus, the conclusion is:
No transformation among the given options matches the criteria to transform the triangle correctly. Therefore, the answer is:
[tex]\[[]\][/tex]
We are provided several options:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]
2. [tex]\(R_{0,180^{\circ}}\)[/tex]
3. [tex]\(R_{0,270^{\circ}}\)[/tex]
4. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
5. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
Step-by-Step Analysis:
1. Test [tex]\(R_{0, 90^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (-y, x)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (-0, 0) = (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (2, 0)\)[/tex] which matches [tex]\(Y^{\prime}(2, 0)\)[/tex].
- [tex]\(Z(-2, 2) \rightarrow (-2, -2)\)[/tex], but [tex]\(Z^{\prime} is (2, -2)\)[/tex], so this transformation does not match.
2. Test [tex]\(R_{0, 180^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (0, 2)\)[/tex] which does not match [tex]\(Y^{\prime}(2, 0)\)[/tex].
- Therefore, this transformation does not match.
3. Test [tex]\(R_{0, 270^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (y, -x)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (-2, 0)\)[/tex] which does not match [tex]\(Y^{\prime}(2, 0)\)[/tex].
- Therefore, this transformation does not match.
Having checked all possible rotations and their coordinate transformation rules, we see that no combination of the listed rotations can transform triangle [tex]\(XYZ\)[/tex] to triangle [tex]\(X^{\prime}Y^{\prime}Z^{\prime}\)[/tex] correctly.
Thus, the conclusion is:
No transformation among the given options matches the criteria to transform the triangle correctly. Therefore, the answer is:
[tex]\[[]\][/tex]
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