Question

Quadrilateral
\[A'B'C'D'\] is the result of reflecting quadrilateral
\[ABCD\] across the line
\[\ell\].
A coordinate plane. The x- and y-axes both scale by one. A quadrilateral A B C D is in the first quadrant. A line of reflection starts at zero, zero and passes through Point C of the quadrilateral.
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[y\]
\[x\]
\[A\]
\[B\]
\[D\]
\[C\]
\[\ell\]
Select all of the correct statements about the unchanged properties of quadrilateral
\[ABCD\] and quadrilateral
\[A'B'C'D'\].
Choose all answers that apply:
Choose all answers that apply:
(Choice A) The perimeters of quadrilateral
\[ABCD\] and quadrilateral
\[A'B'C'D'\] are the same.
A
The perimeters of quadrilateral
\[ABCD\] and quadrilateral
\[A'B'C'D'\] are the same.
(Choice B)

\[\overline{AD}\] and

\[\overline{A'D'}\] have the same lengths.
B

\[\overline{AD}\] and

\[\overline{A'D'}\] have the same lengths.
(Choice C)
\[C\] and
\[C'\] have the same coordinates.
C
\[C\] and
\[C'\] have the same coordinates.
(Choice D) None of the above
D
None of the above

Answer 1

To determine which statements about the quadrilaterals \(ABCD\) and \(A'B'C'D'\) are correct, we must consider the properties of reflection across a line.

1. **Perimeter**: The perimeter of a shape does not change when it is reflected. Thus, the perimeters of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are the same.
- **Choice A** is correct.

2. **Lengths of corresponding sides**: Since reflection preserves distances, the lengths of corresponding sides in \(ABCD\) and \(A'B'C'D'\) will also be the same. Therefore, \(\overline{AD}\) and \(\overline{A'D'}\) have the same lengths.
- **Choice B** is correct.

3. **Coordinates of points**: Reflection does not preserve the coordinates of points. When a point is reflected over a line, its coordinates change, although the distance from the line remains the same. Therefore, \(C\) and \(C'\) (the reflection of \(C\)) will not have the same coordinates.
- **Choice C** is incorrect.

Thus, the correct choices about the unchanged properties of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are:
- **Choice A**: The perimeters of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are the same.
- **Choice B**: \(\overline{AD}\) and \(\overline{A'D'}\) have the same lengths.

So the correct answers are **A** and **B**.