To determine which rotation transformation has been applied to triangle [tex]\( XYZ \)[/tex] resulting in [tex]\( X'Y'Z' \)[/tex], we'll analyze the coordinates of the vertices before and after the rotation.
Initial vertices of triangle [tex]\( XYZ \)[/tex]:
- [tex]\( X = (1, 3) \)[/tex]
- [tex]\( Y = (0, 0) \)[/tex]
- [tex]\( Z = (-1, 2) \)[/tex]
Vertices of the image triangle [tex]\( X'Y'Z' \)[/tex] after the rotation:
- [tex]\( X' = (-3, 1) \)[/tex]
- [tex]\( Y' = (0, 0) \)[/tex]
- [tex]\( Z' = (-2, -1) \)[/tex]
We need to check the coordinates of the vertices after applying different rotation transformations around the origin.
### 1. Rotation by [tex]\( 90^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,90^\circ} \)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by [tex]\( 90^\circ \)[/tex] counterclockwise around the origin is [tex]\((x, y) \to (-y, x)\)[/tex]:
- For [tex]\( X(1, 3) \)[/tex]:
[tex]\[
X \to (-3, 1)
\][/tex]
- For [tex]\( Y(0, 0) \)[/tex]:
[tex]\[
Y \to (0, 0)
\][/tex]
- For [tex]\( Z(-1, 2) \)[/tex]:
[tex]\[
Z \to (-2, -1)
\][/tex]
These calculated points match exactly with the given vertices of [tex]\( X'Y'Z' \)[/tex]:
- [tex]\( X' = (-3, 1) \)[/tex]
- [tex]\( Y' = (0, 0) \)[/tex]
- [tex]\( Z' = (-2, -1) \)[/tex]
Thus, the rotation rule that describes the transformation from [tex]\( XYZ \)[/tex] to [tex]\( X'Y'Z' \)[/tex] is [tex]\( R_{0, 90^\circ} \)[/tex].
Therefore, the correct rule is:
[tex]\[
\boxed{R_{0,90^{\circ}}}
\][/tex]
