To find the rules that describe the rotation from triangle XYZ to triangle X'Y'Z', we can analyze the coordinates of the points in both triangles.
The vertices of triangle XYZ are:
- \(X(-2, 2)\)
- \(Y(1, 2)\)
- \(Z(0, 4)\)
The vertices of triangle X'Y'Z' are:
- \(X'(2, -2)\)
- \(Y'(-1, -1)\)
- \(Z'(0, -4)\)
Analyzing the Points:
-
For Point X to X':
- Original \(X(-2, 2)\) transforms into \(X'(2, -2)\).
- This transformation can be represented as \((x, y) \rightarrow (–y, x)\).
Applying this transformation:
- For \(X(-2, 2)\): \[ (–2, 2) \rightarrow (-2, -2) \text{ which corresponds to } X'(2, -2). \]
-
For Point Y to Y':
- Original \(Y(1, 2)\) transforms into \(Y'(-1, -1)\).
- Applying \((x, y) \rightarrow (–y, x)\): \[ (1, 2) \rightarrow (–2, 1) \text{ which is not equal to } Y'(-1, -1). \]
However, the rotation can involve multiple transformations, and other options need consideration.
Evaluating Rotations:
-
90° Rotation (R0, 90°):
- The points transformed should follow the rule \((x, y) \rightarrow (-y, x)\).
- Evaluating:
- \(X(-2, 2) \rightarrow (-2, -2)\),
- \(Y(1, 2) \rightarrow (-2, 1)\),
- \(Z(0, 4) \rightarrow (-4, 0)\).
-
180° Rotation (R0, 180°):
- The points transformed by \((x, y) \rightarrow (-x, -y)\).
- Evaluating:
- \(X(-2, 2) \rightarrow (2, -2)\),
- \(Y(1, 2) \rightarrow (-1, -2)\),
- \(Z(0, 4) \rightarrow (0, -4)\).
The transformation holds for:
- \(X(-2, 2) \rightarrow (2, -2)\),
- \(Y(1, 2) \rightarrow (-1, -2)\),
- \(Z(0, 4) \rightarrow (0, -4)\) correlating with \(Y'\).
Conclusion:
After analyzing, the rules that could describe the rotation of triangle XYZ to triangle X'Y'Z' are:
- R0, 180°
- (x, y) → (–y, x)
Thus, the selected options are R0, 180° and (x, y) → (–y, x).
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