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The coordinates of the vertices of trapezoid [tex]$ABCD$[/tex] are [tex]$A (2,6), B (5,6), C (7,1)$[/tex], and [tex]$D (-1,1)$[/tex]. The coordinates of the vertices of trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex] are [tex]$A^{\prime}(-6,-2), B^{\prime}(-6,-5), C^{\prime}(-1,-7)$[/tex], and [tex]$D^{\prime}(-1,1)$[/tex].

Which statement correctly describes the relationship between trapezoid [tex]$ABCD$[/tex] and trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex]?

A. Trapezoid [tex]$ABCD$[/tex] is congruent to trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex] because you can map trapezoid [tex]$ABCD$[/tex] to trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/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.

B. Trapezoid [tex]$ABCD$[/tex] is congruent to trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex] because you can map trapezoid [tex]$ABCD$[/tex] to trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex] by rotating it [tex]$180^{\circ}$[/tex] about the origin and then translating it 4 units left, which is a sequence of rigid motions.

C. Trapezoid [tex]$ABCD$[/tex] is not congruent to trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{prine}$[/tex] because there is no sequence of rigid motions that maps trapezoid [tex]$ABCD$[/tex] to trapezoid [tex]$A^{\prime} B^{prine} C^{prine} D^{\prime}$[/tex].

D. Trapezoid [tex]$ABCD$[/tex] is congruent to trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex] because you can map trapezoid [tex]$ABCD$[/tex] to trapezoid [tex]$A^{\prime} B^{\prime} C^{\prime} D^{\prime}$[/tex] by reflecting it across the [tex]$x$[/tex]-axis and then across the [tex]$y$[/tex]-axis, which is a sequence of rigid motions.



Answer :

GinnyAnswer
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]

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