Answer :
To determine which rotational transformation describes the rotation of triangle RST to its image [tex]\(R'S'T'\)[/tex], we can follow these steps:
1. Understand the problem: We need to identify the rotation that maps the points [tex]\(R(2,0)\)[/tex], [tex]\(S(4,0)\)[/tex], and [tex]\(T(1,-3)\)[/tex] to their corresponding points [tex]\(R'(0,-2)\)[/tex], [tex]\(S'(0,-4)\)[/tex], and [tex]\(T'(-3,-1)\)[/tex].
2. Check the transformation rule: We must check for each rotation whether it maps the original points to the new ones.
3. Check rotation 90 degrees counterclockwise: For a rotation by 90 degrees counterclockwise around the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (0,2) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (0,4) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (3,1) \][/tex]
Since [tex]\((2,0)\)[/tex] mapped to [tex]\((0,2)\)[/tex], [tex]\((4,0)\)[/tex] mapped to [tex]\((0,4)\)[/tex], and [tex]\((1,-3)\)[/tex] mapped to [tex]\((3,1)\)[/tex], this rotation does not match the given image.
4. Check rotation 180 degrees counterclockwise: For a rotation by 180 degrees counterclockwise, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (-2,0) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (-4,0) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (-1,3) \][/tex]
Since [tex]\((2,0)\)[/tex] mapped to [tex]\((-2,0)\)[/tex], [tex]\((4,0)\)[/tex] mapped to [tex]\((-4,0)\)[/tex], and [tex]\((1,-3)\)[/tex] mapped to [tex]\((-1,3)\)[/tex], this rotation also does not match the given image.
5. Check rotation 270 degrees counterclockwise: For a rotation by 270 degrees counterclockwise (or 90 degrees clockwise), the transformation rule is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (0,-2) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (0,-4) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (-3,-1) \][/tex]
Since [tex]\((2,0)\)[/tex] mapped to [tex]\((0,-2)\)[/tex], [tex]\((4,0)\)[/tex] mapped to [tex]\((0,-4)\)[/tex], and [tex]\((1,-3)\)[/tex] mapped to [tex]\((-3,-1)\)[/tex], this transformation perfectly matches the given image.
6. Check rotation 360 degrees (or 0 degrees): For a rotation of 360 degrees counterclockwise, the transformation rule is:
[tex]\[ (x, y) \rightarrow (x, y) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (2,0) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (4,0) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (1,-3) \][/tex]
Since the mapping keeps the coordinates unchanged, which does not match the given image.
From the above checks, only the rotation of 270 degrees counterclockwise (or equivalently, 90 degrees clockwise) correctly matches the given transformation. Therefore, the correct rule is:
[tex]\[ R_{0,270} \][/tex]
1. Understand the problem: We need to identify the rotation that maps the points [tex]\(R(2,0)\)[/tex], [tex]\(S(4,0)\)[/tex], and [tex]\(T(1,-3)\)[/tex] to their corresponding points [tex]\(R'(0,-2)\)[/tex], [tex]\(S'(0,-4)\)[/tex], and [tex]\(T'(-3,-1)\)[/tex].
2. Check the transformation rule: We must check for each rotation whether it maps the original points to the new ones.
3. Check rotation 90 degrees counterclockwise: For a rotation by 90 degrees counterclockwise around the origin, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (0,2) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (0,4) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (3,1) \][/tex]
Since [tex]\((2,0)\)[/tex] mapped to [tex]\((0,2)\)[/tex], [tex]\((4,0)\)[/tex] mapped to [tex]\((0,4)\)[/tex], and [tex]\((1,-3)\)[/tex] mapped to [tex]\((3,1)\)[/tex], this rotation does not match the given image.
4. Check rotation 180 degrees counterclockwise: For a rotation by 180 degrees counterclockwise, the transformation rule is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (-2,0) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (-4,0) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (-1,3) \][/tex]
Since [tex]\((2,0)\)[/tex] mapped to [tex]\((-2,0)\)[/tex], [tex]\((4,0)\)[/tex] mapped to [tex]\((-4,0)\)[/tex], and [tex]\((1,-3)\)[/tex] mapped to [tex]\((-1,3)\)[/tex], this rotation also does not match the given image.
5. Check rotation 270 degrees counterclockwise: For a rotation by 270 degrees counterclockwise (or 90 degrees clockwise), the transformation rule is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (0,-2) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (0,-4) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (-3,-1) \][/tex]
Since [tex]\((2,0)\)[/tex] mapped to [tex]\((0,-2)\)[/tex], [tex]\((4,0)\)[/tex] mapped to [tex]\((0,-4)\)[/tex], and [tex]\((1,-3)\)[/tex] mapped to [tex]\((-3,-1)\)[/tex], this transformation perfectly matches the given image.
6. Check rotation 360 degrees (or 0 degrees): For a rotation of 360 degrees counterclockwise, the transformation rule is:
[tex]\[ (x, y) \rightarrow (x, y) \][/tex]
- For [tex]\(R(2,0)\)[/tex]:
[tex]\[ (2,0) \rightarrow (2,0) \][/tex]
- For [tex]\(S(4,0)\)[/tex]:
[tex]\[ (4,0) \rightarrow (4,0) \][/tex]
- For [tex]\(T(1,-3)\)[/tex]:
[tex]\[ (1,-3) \rightarrow (1,-3) \][/tex]
Since the mapping keeps the coordinates unchanged, which does not match the given image.
From the above checks, only the rotation of 270 degrees counterclockwise (or equivalently, 90 degrees clockwise) correctly matches the given transformation. Therefore, the correct rule is:
[tex]\[ R_{0,270} \][/tex]
