Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
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]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.