Question
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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. Which explanation for the congruency of the trapezoids is correct? Statement #1: The trapezoids are congruent because ABCD was reflected over the line y=x . Statement #2: The trapezoids are congruent because ABCD was translated left 4 units and up 2 units. Statement #3: The trapezoids are congruent because ABCD was rotated 270° (counterclockwise). Statement #4: The trapezoids are congruent because ABCD was rotated 90° (counterclockwise). (1 point) Statement # is correct.
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. Which explanation for the congruency of the trapezoids is correct? Statement #1: The trapezoids are congruent because ABCD was reflected over the line y=x . Statement #2: The trapezoids are congruent because ABCD was translated left 4 units and up 2 units. Statement #3: The trapezoids are congruent because ABCD was rotated 270° (counterclockwise). Statement #4: The trapezoids are congruent because ABCD was rotated 90° (counterclockwise). (1 point) Statement # is correct.
Answers
GPT-4o mini
To determine which statement correctly explains the congruency of the trapezoids \(ABCD\) and \(A'B'C'D'\), we have to consider the properties of geometric transformations.
- **Reflection**: A reflection over the line \(y=x\) will change the coordinates of points, but this transformation maintains the shape and size of the figure. Therefore, if the trapezoids are the same shape and size after this transformation, they would be congruent.
- **Translation**: A translation simply moves the entire figure without rotating or flipping it. Thus, if a trapezoid \(ABCD\) is translated left by 4 units and up by 2 units to get \(A'B'C'D'\), these trapezoids are also congruent, as the distance and angles remain unchanged.
- **Rotation (90° and 270°)**: Both rotations will maintain the characteristics of the trapezoid (distances and angles), leading to congruency. A 270° rotation counterclockwise is the same as a 90° rotation clockwise. As with translations, these trapezoids would remain congruent.
Since all statements #1, #2, #3, and #4 describe transformations that preserve congruency, it's essential to choose the one that correctly describes the transformation observed between the given trapezoids.
Therefore, if it is known that one specific transformation (from the statements given) was performed, you would choose that specific statement. If you are unsure of the specific transformation used, you would note that all statements describe transformations that can lead to the congruency of the trapezoids.
If you need to select one answer without additional context to differentiate which transformation was applied specifically, the correct phrasing would be: "All of the given statements could potentially explain congruency depending on the situation."
However, if you only want to pick one correct answer, without external context, please indicate the transformation you wish to base your choice on.
**Let me know if you can specify the transformation that led to the congruency of the trapezoids!**
- **Reflection**: A reflection over the line \(y=x\) will change the coordinates of points, but this transformation maintains the shape and size of the figure. Therefore, if the trapezoids are the same shape and size after this transformation, they would be congruent.
- **Translation**: A translation simply moves the entire figure without rotating or flipping it. Thus, if a trapezoid \(ABCD\) is translated left by 4 units and up by 2 units to get \(A'B'C'D'\), these trapezoids are also congruent, as the distance and angles remain unchanged.
- **Rotation (90° and 270°)**: Both rotations will maintain the characteristics of the trapezoid (distances and angles), leading to congruency. A 270° rotation counterclockwise is the same as a 90° rotation clockwise. As with translations, these trapezoids would remain congruent.
Since all statements #1, #2, #3, and #4 describe transformations that preserve congruency, it's essential to choose the one that correctly describes the transformation observed between the given trapezoids.
Therefore, if it is known that one specific transformation (from the statements given) was performed, you would choose that specific statement. If you are unsure of the specific transformation used, you would note that all statements describe transformations that can lead to the congruency of the trapezoids.
If you need to select one answer without additional context to differentiate which transformation was applied specifically, the correct phrasing would be: "All of the given statements could potentially explain congruency depending on the situation."
However, if you only want to pick one correct answer, without external context, please indicate the transformation you wish to base your choice on.
**Let me know if you can specify the transformation that led to the congruency of the trapezoids!**