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Sagot :
To determine the correct relationship between trapezoid [tex]\(ABCD\)[/tex] and trapezoid [tex]\(A'B'C'D'\)[/tex], we need to analyze the transformations that can map one trapezoid onto the other and see if they preserve congruence.
### Step-by-Step Analysis:
1. Check reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting across the [tex]\(x\)[/tex]-axis changes the [tex]\(y\)[/tex]-coordinates of each point to their negatives while keeping [tex]\(x\)[/tex]-coordinates the same.
- Vertices of [tex]\(ABCD\)[/tex]:
- [tex]\(A (2,6) \rightarrow A'' (2,-6)\)[/tex]
- [tex]\(B (5,6) \rightarrow B'' (5,-6)\)[/tex]
- [tex]\(C (7,1) \rightarrow C'' (7,-1)\)[/tex]
- [tex]\(D (-1,1) \rightarrow D'' (-1,-1)\)[/tex]
2. Check rotation 90° clockwise:
- After reflecting, we need to rotate these new points 90° clockwise. The transformation rule for this is [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
- Vertices after 90° clockwise rotation:
- [tex]\(A'' (2,-6) \rightarrow A'''(-6,-2)\)[/tex]
- [tex]\(B'' (5,-6) \rightarrow B'''(-6,-5)\)[/tex]
- [tex]\(C'' (7,-1) \rightarrow C'''(-1,-7)\)[/tex]
- [tex]\(D'' (-1,-1) \rightarrow D'''(-1,1)\)[/tex]
3. Comparison of final coordinates:
- Compare the coordinates of trapezoid [tex]\(A'''\)[/tex] with trapezoid [tex]\(A'B'C'D'\)[/tex]:
- [tex]\(A'''(-6,-2) = A'(-6,-2)\)[/tex]
- [tex]\(B'''(-6,-5) = B'(-6,-5)\)[/tex]
- [tex]\(C'''(-1,-7) = C'(-1,-7)\)[/tex]
- [tex]\(D'''(-1,1) = D'(-1,1)\)[/tex]
Since all map exactly using the transformations mentioned, we see that this sequence of transformations correctly maps [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex].
Therefore, trapezoid [tex]\(ABCD\)[/tex] is congruent to trapezoid [tex]\(A'B'C'D'\)[/tex] because you can map trapezoid [tex]\(ABCD\)[/tex] to trapezoid [tex]\(A'B'C'D'\)[/tex] by reflecting it across the [tex]\(x\)[/tex]-axis and then rotating it [tex]\(90^\circ\)[/tex] clockwise, which is a sequence of rigid motions.
Thus, the correct statement is:
[tex]\[ \boxed{\text{Trapezoid } ABCD \text{ is congruent to trapezoid } A'B'C'D' \text{ because you can map trapezoid } ABCD \text{ to trapezoid } A'B'C'D' \text{ by reflecting it across the } x\text{-axis and then rotating it } 90^\circ \text{ clockwise, which is a sequence of rigid motions.}} \][/tex]
### Step-by-Step Analysis:
1. Check reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting across the [tex]\(x\)[/tex]-axis changes the [tex]\(y\)[/tex]-coordinates of each point to their negatives while keeping [tex]\(x\)[/tex]-coordinates the same.
- Vertices of [tex]\(ABCD\)[/tex]:
- [tex]\(A (2,6) \rightarrow A'' (2,-6)\)[/tex]
- [tex]\(B (5,6) \rightarrow B'' (5,-6)\)[/tex]
- [tex]\(C (7,1) \rightarrow C'' (7,-1)\)[/tex]
- [tex]\(D (-1,1) \rightarrow D'' (-1,-1)\)[/tex]
2. Check rotation 90° clockwise:
- After reflecting, we need to rotate these new points 90° clockwise. The transformation rule for this is [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
- Vertices after 90° clockwise rotation:
- [tex]\(A'' (2,-6) \rightarrow A'''(-6,-2)\)[/tex]
- [tex]\(B'' (5,-6) \rightarrow B'''(-6,-5)\)[/tex]
- [tex]\(C'' (7,-1) \rightarrow C'''(-1,-7)\)[/tex]
- [tex]\(D'' (-1,-1) \rightarrow D'''(-1,1)\)[/tex]
3. Comparison of final coordinates:
- Compare the coordinates of trapezoid [tex]\(A'''\)[/tex] with trapezoid [tex]\(A'B'C'D'\)[/tex]:
- [tex]\(A'''(-6,-2) = A'(-6,-2)\)[/tex]
- [tex]\(B'''(-6,-5) = B'(-6,-5)\)[/tex]
- [tex]\(C'''(-1,-7) = C'(-1,-7)\)[/tex]
- [tex]\(D'''(-1,1) = D'(-1,1)\)[/tex]
Since all map exactly using the transformations mentioned, we see that this sequence of transformations correctly maps [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex].
Therefore, trapezoid [tex]\(ABCD\)[/tex] is congruent to trapezoid [tex]\(A'B'C'D'\)[/tex] because you can map trapezoid [tex]\(ABCD\)[/tex] to trapezoid [tex]\(A'B'C'D'\)[/tex] by reflecting it across the [tex]\(x\)[/tex]-axis and then rotating it [tex]\(90^\circ\)[/tex] clockwise, which is a sequence of rigid motions.
Thus, the correct statement is:
[tex]\[ \boxed{\text{Trapezoid } ABCD \text{ is congruent to trapezoid } A'B'C'D' \text{ because you can map trapezoid } ABCD \text{ to trapezoid } A'B'C'D' \text{ by reflecting it across the } x\text{-axis and then rotating it } 90^\circ \text{ clockwise, which is a sequence of rigid motions.}} \][/tex]
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