Question
Two trapezoids, upper A upper B upper C upper D and upper A prime upper B prime upper C prime upper D prime, are graphed on a coordinate plane.
Samantha determined that trapezoid ABCD≅A′B′C′D′
. Which of the following should be her next step?
Statement #1: Verify that ABCD≅A′B′C′D′
using the mapping equation for the translation A(x,y)→A(x+8,y)
.
Statement #2: Verify that ABCD≅A′B′C′D′
using the mapping equation for a reflection over the x-axis A(x,y)→A(x,−y)
.
Statement #3: Verify that ABCD≅A′B′C′D′
using the mapping equation for a reflection over the y-axis A(x,y)→A(−x,y)
.
Samantha determined that trapezoid ABCD≅A′B′C′D′
. Which of the following should be her next step?
Statement #1: Verify that ABCD≅A′B′C′D′
using the mapping equation for the translation A(x,y)→A(x+8,y)
.
Statement #2: Verify that ABCD≅A′B′C′D′
using the mapping equation for a reflection over the x-axis A(x,y)→A(x,−y)
.
Statement #3: Verify that ABCD≅A′B′C′D′
using the mapping equation for a reflection over the y-axis A(x,y)→A(−x,y)
.
Answers
GPT-4o mini
To verify that two trapezoids, \( ABCD \) and \( A'B'C'D' \), are congruent (denoted as \( ABCD \cong A'B'C'D' \)), Samantha should determine whether one trapezoid can be obtained from the other through a series of transformations, such as translations, reflections, or rotations.
Since she has already determined that the trapezoids are congruent, her next step would typically involve verifying the specific type of transformation that relates the two shapes. Let's assess each of the statements to see which would offer the most relevant verification step.
1. **Statement #1: Verify that \( ABCD \cong A'B'C'D' \) using the mapping equation for the translation \( A(x,y) \rightarrow A(x+8,y) \)**.
- This statement suggests verifying the congruence under a specific translation. If \( A'B'C'D' \) is a translation of \( ABCD \) by 8 units to the right, it would establish the congruence.
2. **Statement #2: Verify that \( ABCD \cong A'B'C'D' \) using the mapping equation for a reflection over the x-axis \( A(x,y) \rightarrow A(x,-y) \)**.
- This reflection would show that \( A'B'C'D' \) could be obtained by flipping \( ABCD \) over the x-axis. While this transformation could show congruence, it may not be the correct next step if translation is what has occurred.
3. **Statement #3: Verify that \( ABCD \cong A'B'C'D' \) using the mapping equation for a reflection over the y-axis \( A(x,y) \rightarrow A(-x,y) \)**.
- Similar to Statement #2, this reflection shows a different transformation but may not be the specific transformation that was applied.
Given these options, if the translation has been determined to relate the two trapezoids, then **Statement #1** is appropriate as a next step. If any other transformation (like reflection) applies and has already been established, then **Statement #2** or **Statement #3** might be relevant.
Thus, without additional context for whether a translation or reflection is confirmed, **Statement #1** would be her best next step if a translation was already involved in her initial analysis.
Since she has already determined that the trapezoids are congruent, her next step would typically involve verifying the specific type of transformation that relates the two shapes. Let's assess each of the statements to see which would offer the most relevant verification step.
1. **Statement #1: Verify that \( ABCD \cong A'B'C'D' \) using the mapping equation for the translation \( A(x,y) \rightarrow A(x+8,y) \)**.
- This statement suggests verifying the congruence under a specific translation. If \( A'B'C'D' \) is a translation of \( ABCD \) by 8 units to the right, it would establish the congruence.
2. **Statement #2: Verify that \( ABCD \cong A'B'C'D' \) using the mapping equation for a reflection over the x-axis \( A(x,y) \rightarrow A(x,-y) \)**.
- This reflection would show that \( A'B'C'D' \) could be obtained by flipping \( ABCD \) over the x-axis. While this transformation could show congruence, it may not be the correct next step if translation is what has occurred.
3. **Statement #3: Verify that \( ABCD \cong A'B'C'D' \) using the mapping equation for a reflection over the y-axis \( A(x,y) \rightarrow A(-x,y) \)**.
- Similar to Statement #2, this reflection shows a different transformation but may not be the specific transformation that was applied.
Given these options, if the translation has been determined to relate the two trapezoids, then **Statement #1** is appropriate as a next step. If any other transformation (like reflection) applies and has already been established, then **Statement #2** or **Statement #3** might be relevant.
Thus, without additional context for whether a translation or reflection is confirmed, **Statement #1** would be her best next step if a translation was already involved in her initial analysis.