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.