Answer :
To find the coordinates of the vertices of the pre-image trapezoid [tex]\( ABCD \)[/tex] that were mapped to the final image [tex]\( A"B"C"D" \)[/tex] by the transformation [tex]\((r_{r.1} \circ T_{4, 0})(x, y)\)[/tex], we need to reverse the transformation step by step.
The given transformation rule is a combination of two steps:
1. A rotation [tex]\( r_{r.1} \)[/tex] which rotates the point 90 degrees counter-clockwise.
2. A translation [tex]\( T_{4,0} \)[/tex] which translates the point by the vector (4, 0).
### Step-by-Step Explanation:
1. Define the transformations:
- Rotation [tex]\( r_{r.1} \)[/tex]:
Rotation by 90 degrees counter-clockwise can be represented as:
[tex]\[ R(x, y) = (-y, x) \][/tex]
- Translation [tex]\( T_{4,0} \)[/tex]:
Translation by the vector (4, 0) can be represented as:
[tex]\[ T_{4,0}(x, y) = (x + 4, y) \][/tex]
2. Combined Transformation:
Apply the rotation first and then the translation:
[tex]\[ (r_{r.1} \circ T_{4,0})(x, y) = T_{4,0}(R(x, y)) = T_{4,0}(-y, x) \][/tex]
This gives:
[tex]\[ (r_{r.1} \circ T_{4,0})(x, y) = (-y + 4, x) \][/tex]
3. Finding the Pre-image:
We need to determine which of the provided coordinate pairs, when transformed, match the coordinates of the vertices in the final image [tex]\( A"B"C"D" \)[/tex]. According to the problem, the transformed vertices are [tex]\( (3, 4), (9, 4), (9, -1), (3, -1) \)[/tex].
Let's check each option:
- For the pair [tex]\((-1, 0)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(-1, 0) = (0 + 4, -1) = (4, -1) \][/tex]
Not matching any final vertices.
- For the pair [tex]\((-1, -5)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(-1, -5) = (5 + 4, -1) = (9, -1) \][/tex]
Matching the final vertex [tex]\( (9, -1) \)[/tex].
- For the pair [tex]\((1, 1)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(1, 1) = (-1 + 4, 1) = (3, 1) \][/tex]
Not matching any final vertices.
- For the pair [tex]\((7, 0)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(7, 0) = (0 + 4, 7) = (4, 7) \][/tex]
Not matching any final vertices.
- For the pair [tex]\((7, -5)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(7, -5) = (5 + 4, 7) = (9, 7) \][/tex]
Not matching any final vertices.
The only pair that matches one of the final vertices [tex]\( (9, -1) \)[/tex] is [tex]\((-1, -5)\)[/tex].
### Conclusion:
The ordered pair that names the coordinates of the vertices of the pre-image trapezoid [tex]\( ABCD \)[/tex] is:
[tex]\[ (-1, -5) \][/tex]
We should select this option as the pre-image vertex (out of the provided coordinate options) that produced a vertex of the final image [tex]\( A"B"C"D" \)[/tex].
The given transformation rule is a combination of two steps:
1. A rotation [tex]\( r_{r.1} \)[/tex] which rotates the point 90 degrees counter-clockwise.
2. A translation [tex]\( T_{4,0} \)[/tex] which translates the point by the vector (4, 0).
### Step-by-Step Explanation:
1. Define the transformations:
- Rotation [tex]\( r_{r.1} \)[/tex]:
Rotation by 90 degrees counter-clockwise can be represented as:
[tex]\[ R(x, y) = (-y, x) \][/tex]
- Translation [tex]\( T_{4,0} \)[/tex]:
Translation by the vector (4, 0) can be represented as:
[tex]\[ T_{4,0}(x, y) = (x + 4, y) \][/tex]
2. Combined Transformation:
Apply the rotation first and then the translation:
[tex]\[ (r_{r.1} \circ T_{4,0})(x, y) = T_{4,0}(R(x, y)) = T_{4,0}(-y, x) \][/tex]
This gives:
[tex]\[ (r_{r.1} \circ T_{4,0})(x, y) = (-y + 4, x) \][/tex]
3. Finding the Pre-image:
We need to determine which of the provided coordinate pairs, when transformed, match the coordinates of the vertices in the final image [tex]\( A"B"C"D" \)[/tex]. According to the problem, the transformed vertices are [tex]\( (3, 4), (9, 4), (9, -1), (3, -1) \)[/tex].
Let's check each option:
- For the pair [tex]\((-1, 0)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(-1, 0) = (0 + 4, -1) = (4, -1) \][/tex]
Not matching any final vertices.
- For the pair [tex]\((-1, -5)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(-1, -5) = (5 + 4, -1) = (9, -1) \][/tex]
Matching the final vertex [tex]\( (9, -1) \)[/tex].
- For the pair [tex]\((1, 1)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(1, 1) = (-1 + 4, 1) = (3, 1) \][/tex]
Not matching any final vertices.
- For the pair [tex]\((7, 0)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(7, 0) = (0 + 4, 7) = (4, 7) \][/tex]
Not matching any final vertices.
- For the pair [tex]\((7, -5)\)[/tex]:
[tex]\[ (r_{r.1} \circ T_{4, 0})(7, -5) = (5 + 4, 7) = (9, 7) \][/tex]
Not matching any final vertices.
The only pair that matches one of the final vertices [tex]\( (9, -1) \)[/tex] is [tex]\((-1, -5)\)[/tex].
### Conclusion:
The ordered pair that names the coordinates of the vertices of the pre-image trapezoid [tex]\( ABCD \)[/tex] is:
[tex]\[ (-1, -5) \][/tex]
We should select this option as the pre-image vertex (out of the provided coordinate options) that produced a vertex of the final image [tex]\( A"B"C"D" \)[/tex].
