Answer :
To determine which rule describes the transformation of a point [tex]\((x, y)\)[/tex] when a triangle is rotated [tex]\(90^{\circ}\)[/tex] about the origin, we can analyze the effect of a [tex]\(90^{\circ}\)[/tex] rotation in general terms:
1. Understand the standard rotation rules:
- A [tex]\(90^{\circ}\)[/tex] rotation counterclockwise around the origin changes a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- A [tex]\(90^{\circ}\)[/tex] rotation clockwise around the origin changes a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
2. Confirm the behavior for specific points:
- Starting with point [tex]\( (1, 0) \)[/tex]:
- After a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation, it becomes [tex]\((0, 1)\)[/tex].
- Check the rules provided to see which matches this behavior:
- [tex]\((1, 0) \rightarrow (-0, 1) \rightarrow (0, 1)\)[/tex] fits with [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
- start with point [tex]\((0, 1)\)[/tex]
- After a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation, it becomes [tex]\((-1, 0)\)[/tex].
- Check the rules provided:
- [tex]\((0, 1) \rightarrow (-1, 0)\)[/tex] fits with [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
3. Given this analysis, we see that [tex]\((x, y) \rightarrow (-y, x)\)[/tex] matches the transformation for a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation about the origin.
Therefore, the correct rule describing the transformation is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
1. Understand the standard rotation rules:
- A [tex]\(90^{\circ}\)[/tex] rotation counterclockwise around the origin changes a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- A [tex]\(90^{\circ}\)[/tex] rotation clockwise around the origin changes a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
2. Confirm the behavior for specific points:
- Starting with point [tex]\( (1, 0) \)[/tex]:
- After a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation, it becomes [tex]\((0, 1)\)[/tex].
- Check the rules provided to see which matches this behavior:
- [tex]\((1, 0) \rightarrow (-0, 1) \rightarrow (0, 1)\)[/tex] fits with [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
- start with point [tex]\((0, 1)\)[/tex]
- After a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation, it becomes [tex]\((-1, 0)\)[/tex].
- Check the rules provided:
- [tex]\((0, 1) \rightarrow (-1, 0)\)[/tex] fits with [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
3. Given this analysis, we see that [tex]\((x, y) \rightarrow (-y, x)\)[/tex] matches the transformation for a [tex]\(90^{\circ}\)[/tex] counterclockwise rotation about the origin.
Therefore, the correct rule describing the transformation is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
