Triangle XYZ is rotated to create the image triangle X'Y'Z'.

On a coordinate plane, 2 triangles are shown. The first triangle has points X (negative 2, 2), Y (1, 2), Z (0, 4). The second triangle has points X prime (2, negative 2), Y prime (negative 1, negative 1), Z prime (0, negative 4).
Which rules could describe the rotation? Select two options.

R0, 90°
R0, 180°
R0, 270°
(x, y) → (–y, x)
(x, y) → (–x, –y)

To determine which rules describe the rotation of triangle XYZ to its image triangle X'Y'Z', let's analyze the given points.

Original triangle XYZ:
- \(X(-2, 2)\)
- \(Y(1, 2)\)
- \(Z(0, 4)\)

Image triangle \(X'Y'Z'\):
- \(X'(2, -2)\)
- \(Y'(-1, -1)\)
- \(Z'(0, -4)\)

To find the rotation, let's apply the given transformations to the original points and see which one matches the image points.

### Checking each option:

1. **\(R_0, 90°\) rotation:**
   - Formula: \((x, y) \rightarrow (-y, x)\)
   - \(X(-2, 2) \rightarrow (-2, -2)\) (This does not match \(X'(2, -2)\))
   - \(Y(1, 2) \rightarrow (-2, 1)\) (This does not match \(Y'(-1, -1)\))
   - \(Z(0, 4) \rightarrow (-4, 0)\) (This does not match \(Z'(0, -4)\))

   **Conclusion: This option is incorrect.**

2. **\(R_0, 180°\) rotation:**
   - Formula: \((x, y) \rightarrow (-x, -y)\)
   - \(X(-2, 2) \rightarrow (2, -2)\) (This matches \(X'(2, -2)\))
   - \(Y(1, 2) \rightarrow (-1, -2)\) (This does not match \(Y'(-1, -1)\))
   - \(Z(0, 4) \rightarrow (0, -4)\) (This matches \(Z'(0, -4)\))

   **Conclusion: This option is partially correct.**

3. **\(R_0, 270°\) rotation:**
   - Formula: \((x, y) \rightarrow (y, -x)\)
   - \(X(-2, 2) \rightarrow (2, 2)\) (This does not match \(X'(2, -2)\))
   - \(Y(1, 2) \rightarrow (2, -1)\) (This does not match \(Y'(-1, -1)\))
   - \(Z(0, 4) \rightarrow (4, 0)\) (This does not match \(Z'(0, -4)\))

   **Conclusion: This option is incorrect.**

4. **\((x, y) \rightarrow (-y, x)\) transformation:**
   - \(X(-2, 2) \rightarrow (-2, -2)\) (This does not match \(X'(2, -2)\))
   - \(Y(1, 2) \rightarrow (-2, 1)\) (This does not match \(Y'(-1, -1)\))
   - \(Z(0, 4) \rightarrow (-4, 0)\) (This does not match \(Z'(0, -4)\))

   **Conclusion: This option is incorrect.**

5. **\((x, y) \rightarrow (-x, -y)\) transformation:**
   - \(X(-2, 2) \rightarrow (2, -2)\) (This matches \(X'(2, -2)\))
   - \(Y(1, 2) \rightarrow (-1, -2)\) (This does not match \(Y'(-1, -1)\))
   - \(Z(0, 4) \rightarrow (0, -4)\) (This matches \(Z'(0, -4)\))

   **Conclusion: This option is partially correct.**

From the analysis, the two correct options that describe the rotation are:
- \(R_0, 180°\)
- \((x, y) \rightarrow (-x, -y)\)

Both of these options describe the same rotation of 180 degrees about the origin.

Would you like more details or have any questions?

### Related Questions:

1. What are the coordinates of a point after a \(90°\) rotation about the origin?
2. How does a \(270°\) rotation about the origin affect the coordinates of a point?
3. Can you provide an example of a different set of points and apply a \(180°\) rotation to them?
4. How does the transformation \((x, y) \rightarrow (y, -x)\) affect a point's coordinates?
5. What are the general rules for rotating points about the origin in a coordinate plane?

### Tip:
When rotating a point around the origin, remember that a \(90°\) rotation counterclockwise swaps the coordinates and changes the sign of the new x-coordinate, \(180°\) rotation changes both coordinates' signs, and \(270°\) rotation counterclockwise swaps the coordinates and changes the sign of the new y-coordinate.

Triangle XYZ is rotated to create the image triangle X'Y'Z'. On a coordinate plane, 2 triangles are shown. The first triangle has points X (negative 2, 2), Y (1, 2), Z (0, 4). The second triangle has points X prime (2, negative 2), Y prime (negative 1, negative 1), Z prime (0, negative 4). Which rules could describe the rotation? Select two options. R0, 90°, R0, 180°, R0, 270°, (x, y) → (–y, x), (x, y) → (–x, –y).

Explore how to determine the correct rotation rules for triangle XYZ and its image X'Y'Z' on a coordinate plane. Learn about rotation formulas such as 90°, 180°, and 270° transformations, and find out how these rules apply to given coordinates. Suitable for students in Grades 8-10.

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