Let's analyze the rotation of triangle [tex]\(XYZ\)[/tex] with vertices [tex]\(X(0,0)\)[/tex], [tex]\(Y(0,-2)\)[/tex], and [tex]\(Z(-2,-2)\)[/tex] to the image triangle [tex]\(X^{\prime}(0,0)\)[/tex], [tex]\(Y^{\prime}(2,0)\)[/tex], and [tex]\(Z^{\prime}(2,-2)\)[/tex].
Option 1: [tex]\(R_{0.9}, 90^{\circ}\)[/tex]
This option refers to some unconventional parameter [tex]\(0.9\)[/tex], which is unusual and does not standardly describe a normal rotation of 90 degrees. Thus, this rule does not seem correct.
Option 2: [tex]\(R_{0, 180^{\circ}}\)[/tex]
A [tex]\(180^{\circ}\)[/tex] rotation about the origin would transform [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex]. Applying this to the vertices of the original triangle:
- [tex]\(X(0,0)\)[/tex] stays [tex]\((0,0)\)[/tex]
- [tex]\(Y(0,-2)\)[/tex] becomes [tex]\(Y'(0,2)\)[/tex]
- [tex]\(Z(-2,-2)\)[/tex] becomes [tex]\(Z'(2,2)\)[/tex]
These don't match the coordinates of the image triangle, so this rule is not correct.
Option 3: [tex]\(R_0, 270^{\circ}\)[/tex]
A [tex]\(270^{\circ}\)[/tex] rotation (or [tex]\(-90^{\circ}\)[/tex]) about the origin will transform [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex]. Applying this to the vertices:
- [tex]\(X(0,0)\)[/tex] becomes [tex]\((0,0)\)[/tex]
- [tex]\(Y(0,-2)\)[/tex] becomes [tex]\((2,0)\)[/tex]
- [tex]\(Z(-2,-2)\)[/tex] becomes [tex]\((2,-2)\)[/tex]
These match the coordinates of the image triangle exactly, so this is a valid rule.
Option 4: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
This rule represents a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation, which transforms [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex]. Applying this:
- [tex]\(X(0,0)\)[/tex] becomes [tex]\((0,0)\)[/tex]
- [tex]\(Y(0,-2)\)[/tex] becomes [tex]\((2,0)\)[/tex]
- [tex]\(Z(-2,-2)\)[/tex] becomes [tex]\((2,-2)\)[/tex]
These also match the coordinates of the image triangle exactly, so this is another valid rule.
Option 5: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
This notation suggests transforming similarly to a [tex]\(270^{\circ}\)[/tex] rotation but in an unusual format. Let’s apply:
- [tex]\(X(0,0)\)[/tex] would go to [tex]\((0,0)\)[/tex]
- [tex]\(Y(0,-2)\)[/tex] would go to [tex]\((-2,0)\)[/tex]
- [tex]\(Z(-2,-2)\)[/tex] would go to [tex]\((-2,2)\)[/tex]
These results do not fit the image coordinates, so this rule is also incorrect.
Thus, the correct rules that describe the rotation are:
1. [tex]\( R_0, 270^{\circ} \)[/tex]
2. [tex]\( (x, y) \rightarrow (-y, x) \)[/tex]
The selected options are:
[tex]$ R_{0, 270^{\circ}} $[/tex]
[tex]$ (x, y) \rightarrow (-y, x) $[/tex]
