Let's analyze the given vertices of the original triangle and their corresponding vertices after the transformation to identify the transformation rule.
Original coordinates of the triangle vertices:
- [tex]\( X(1,3) \)[/tex]
- [tex]\( Y(0,0) \)[/tex]
- [tex]\( Z(-1,2) \)[/tex]
Rotated coordinates of the triangle vertices:
- [tex]\( X^{\prime}(-3,1) \)[/tex]
- [tex]\( Y^{\prime}(0,0) \)[/tex]
- [tex]\( Z^{\prime}(-2,-1) \)[/tex]
To identify the transformation, we need to determine how each original point maps to its new position.
1. Checking [tex]\( X(1, 3) \rightarrow X^{\prime}(-3, 1) \)[/tex]:
- A 90-degree counter-clockwise rotation around the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- For [tex]\( X(1, 3) \)[/tex]:
[tex]\[
(1, 3) \rightarrow (-3, 1)
\][/tex]
- This matches [tex]\( X^{\prime}(-3, 1) \)[/tex].
2. Checking [tex]\( Y(0, 0) \rightarrow Y^{\prime}(0, 0) \)[/tex]:
- The origin remains unchanged under any rotation.
- For [tex]\( Y(0, 0) \)[/tex]:
[tex]\[
(0, 0) \rightarrow (0, 0)
\][/tex]
- This matches [tex]\( Y^{\prime}(0, 0) \)[/tex].
3. Checking [tex]\( Z(-1, 2) \rightarrow Z^{\prime}(-2, -1) \)[/tex]:
- For [tex]\( Z(-1, 2) \)[/tex] under the same 90-degree counter-clockwise rotation:
[tex]\[
(-1, 2) \rightarrow (-2, -1)
\][/tex]
- This matches [tex]\( Z^{\prime}(-2, -1) \)[/tex].
Since the image of each vertex corresponds correctly with a 90-degree counter-clockwise rotation, we can conclude that the transformation rule is:
[tex]\[
R_{0,90^{\circ}}
\][/tex]
Therefore, the rule that describes the transformation is [tex]\( \boxed{R_{0,90^{\circ}}} \)[/tex].
