To determine which rotation was applied to the triangle [tex]\( XYZ \)[/tex] to obtain [tex]\( X'Y'Z' \)[/tex], we need to analyze the given vertices before and after the transformation.
1. Vertices of the original triangle [tex]\( XYZ \)[/tex]:
[tex]\[
X(1, 3), \quad Y(0, 0), \quad Z(-1, 2)
\][/tex]
2. Vertices of the transformed triangle [tex]\( X'Y'Z' \)[/tex]:
[tex]\[
X'(-3, 1), \quad Y'(0, 0), \quad Z'(-2, -1)
\][/tex]
Next, let's evaluate the potential rotations individually and see which one matches the given transformed vertices. We'll start by knowing typical rotation rules around the origin (counterclockwise):
### 1. Rotation by [tex]\( 90^\circ \)[/tex]
The transformation rule for a rotation of [tex]\( 90^\circ \)[/tex] counterclockwise around the origin is:
[tex]\[
(x, y) \rightarrow (-y, x)
\][/tex]
- For [tex]\( X(1, 3) \)[/tex]:
[tex]\[
(1, 3) \rightarrow (-3, 1)
\][/tex]
This matches [tex]\( X'(-3, 1) \)[/tex].
- For [tex]\( Y(0, 0) \)[/tex]:
[tex]\[
(0, 0) \rightarrow (0, 0)
\][/tex]
This matches [tex]\( Y'(0, 0) \)[/tex].
- For [tex]\( Z(-1, 2) \)[/tex]:
[tex]\[
(-1, 2) \rightarrow (-2, -1)
\][/tex]
This matches [tex]\( Z'(-2, -1) \)[/tex].
Since all the vertices match, the rule that describes the transformation is:
[tex]\[
R_{0, 270^\circ}
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{R_{0,270^{\circ}}}
\][/tex]
