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To determine which rule describes the transformation of triangle RST to its image after rotation, we need to find the angle of rotation that maps the original vertices [tex]\( R(2,0) \)[/tex], [tex]\( S(4,0) \)[/tex], and [tex]\( T(1,-3) \)[/tex] to the transformed vertices [tex]\( R'(0,-2) \)[/tex], [tex]\( S'(0,-4) \)[/tex], and [tex]\( T'(-3,-1) \)[/tex].
Rotation transformations about the origin can commonly occur at angles of 90°, 180°, 270°, or 360°. We will check each of these rotations to see which one matches our transformation.
Step-by-Step Solution:
1. Rotation by 90°:
- Rotation by 90° counterclockwise interchanges coordinates and negates the y-coordinate of the original point.
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- Calculating for vertices:
- [tex]\( R(2,0) \rightarrow (0, 2) \)[/tex]
- [tex]\( S(4,0) \rightarrow (0, 4) \)[/tex]
- [tex]\( T(1,-3) \rightarrow (3, 1) \)[/tex]
- These points do not match [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
2. Rotation by 180°:
- Rotation by 180° negates both coordinates.
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
- Calculating for vertices:
- [tex]\( R(2,0) \rightarrow (-2, 0) \)[/tex]
- [tex]\( S(4,0) \rightarrow (-4, 0) \)[/tex]
- [tex]\( T(1,-3) \rightarrow (-1, 3) \)[/tex]
- These points do not match [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
3. Rotation by 270°:
- Rotation by 270° (or -90°) counterclockwise interchanges coordinates and negates the x-coordinate of the original point.
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
- Calculating for vertices:
- [tex]\( R(2,0) \rightarrow (0, -2) \)[/tex]
- [tex]\( S(4,0) \rightarrow (0, -4) \)[/tex]
- [tex]\( T(1,-3) \rightarrow (-3, -1) \)[/tex]
- These points match exactly with [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
4. Rotation by 360°:
- Rotation by 360° (a full circle) brings points back to their original position.
[tex]\[ (x, y) \rightarrow (x, y) \][/tex]
- So the points would not change and clearly does not match [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
Thus, the correct transformation that maps the original vertices to the image vertices is a rotation of 270° counterclockwise.
Therefore, the rule that describes the transformation is: [tex]\( R_{0, 270^\circ} \)[/tex].
Rotation transformations about the origin can commonly occur at angles of 90°, 180°, 270°, or 360°. We will check each of these rotations to see which one matches our transformation.
Step-by-Step Solution:
1. Rotation by 90°:
- Rotation by 90° counterclockwise interchanges coordinates and negates the y-coordinate of the original point.
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- Calculating for vertices:
- [tex]\( R(2,0) \rightarrow (0, 2) \)[/tex]
- [tex]\( S(4,0) \rightarrow (0, 4) \)[/tex]
- [tex]\( T(1,-3) \rightarrow (3, 1) \)[/tex]
- These points do not match [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
2. Rotation by 180°:
- Rotation by 180° negates both coordinates.
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
- Calculating for vertices:
- [tex]\( R(2,0) \rightarrow (-2, 0) \)[/tex]
- [tex]\( S(4,0) \rightarrow (-4, 0) \)[/tex]
- [tex]\( T(1,-3) \rightarrow (-1, 3) \)[/tex]
- These points do not match [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
3. Rotation by 270°:
- Rotation by 270° (or -90°) counterclockwise interchanges coordinates and negates the x-coordinate of the original point.
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
- Calculating for vertices:
- [tex]\( R(2,0) \rightarrow (0, -2) \)[/tex]
- [tex]\( S(4,0) \rightarrow (0, -4) \)[/tex]
- [tex]\( T(1,-3) \rightarrow (-3, -1) \)[/tex]
- These points match exactly with [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
4. Rotation by 360°:
- Rotation by 360° (a full circle) brings points back to their original position.
[tex]\[ (x, y) \rightarrow (x, y) \][/tex]
- So the points would not change and clearly does not match [tex]\( R'(0, -2) \)[/tex], [tex]\( S'(0, -4) \)[/tex], [tex]\( T'(-3, -1) \)[/tex].
Thus, the correct transformation that maps the original vertices to the image vertices is a rotation of 270° counterclockwise.
Therefore, the rule that describes the transformation is: [tex]\( R_{0, 270^\circ} \)[/tex].
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