Question
Two triangles, upper A upper B upper C and upper D upper E upper F, are graphed on a coordinate plane. The x-axis ranges from negative 5 to 6 in increments of 1. The y-axis ranges from negative 3 to 5 in increments of 1.
Determine the congruence transformation used on △ABC that resulted in △DEF .
(1 point)
Responses
△ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up.
triangle upper A upper B upper C was reflected across the y -axis, then shifted 2 units to the right and 2 units up.
△ABC was shifted 2 units to the right and 2 units up, then rotated 180° (counterclockwise).
triangle upper A upper B upper C was shifted 2 units to the right and 2 units up, then rotated 180 degrees (counterclockwise).
△ABC was rotated −180° (clockwise) or 180° (counterclockwise), then shifted 2 units to the right and 2 units up.
triangle upper A upper B upper C was rotated negative 180 degrees (clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up.
△ABC was shifted 2 units to the left and 2 units up, then reflected across the y-axis.
Determine the congruence transformation used on △ABC that resulted in △DEF .
(1 point)
Responses
△ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up.
triangle upper A upper B upper C was reflected across the y -axis, then shifted 2 units to the right and 2 units up.
△ABC was shifted 2 units to the right and 2 units up, then rotated 180° (counterclockwise).
triangle upper A upper B upper C was shifted 2 units to the right and 2 units up, then rotated 180 degrees (counterclockwise).
△ABC was rotated −180° (clockwise) or 180° (counterclockwise), then shifted 2 units to the right and 2 units up.
triangle upper A upper B upper C was rotated negative 180 degrees (clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up.
△ABC was shifted 2 units to the left and 2 units up, then reflected across the y-axis.
Answers
GPT-4o mini
To determine the correct congruence transformation that transforms triangle \(ABC\) into triangle \(DEF\), we need to analyze the transformations listed in the options.
1. **Reflection across the y-axis:** This transformation flips triangle \(ABC\) over the y-axis. If any vertices of triangle \(ABC\) have positive x-coordinates, after reflecting, those coordinates will become negative and vice versa.
2. **Shifting 2 units to the right and 2 units up:** This transformation increases the x-coordinates of all vertices by 2 and the y-coordinates by 2.
3. **Rotation:** A rotation of 180 degrees (in any direction) effectively turns the triangle upside down around the center point.
4. **Shifting 2 units to the left:** This decreases the x-coordinates of all vertices by 2 while keeping y-coordinates the same.
To identify the correct transformation, consider examining the coordinates of triangles \(ABC\) and \(DEF\). However, if the coordinates are not provided here, it's crucial to reason through the transformations based on the general effects they have. For example:
- If triangle \(DEF\) has its vertices moved from \(ABC\), and it appears directly above/below or laterally aligned with \(ABC\), this may indicate a rotational or reflection move followed by a translation.
- Analyzing shifts either to the right, left, up, or down will provide significant insight into the nature of the transformation.
Based on typical transformations:
The correct choice is usually characterized by a transformation that maintains rigid motion properties (i.e., the shape and size of the triangle remain constant).
Assuming triangle \(ABC\) has been transformed into triangle \(DEF\) through reflection and translation (and without loss of generality, without coordinates), the answer that seems most likely plausible based on a straightforward analysis is:
**The first option:** "△ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up."
If you have specific coordinates, substituting those will give you precise confirmation. Otherwise, do these aspects guide your analysis of the transformation?
1. **Reflection across the y-axis:** This transformation flips triangle \(ABC\) over the y-axis. If any vertices of triangle \(ABC\) have positive x-coordinates, after reflecting, those coordinates will become negative and vice versa.
2. **Shifting 2 units to the right and 2 units up:** This transformation increases the x-coordinates of all vertices by 2 and the y-coordinates by 2.
3. **Rotation:** A rotation of 180 degrees (in any direction) effectively turns the triangle upside down around the center point.
4. **Shifting 2 units to the left:** This decreases the x-coordinates of all vertices by 2 while keeping y-coordinates the same.
To identify the correct transformation, consider examining the coordinates of triangles \(ABC\) and \(DEF\). However, if the coordinates are not provided here, it's crucial to reason through the transformations based on the general effects they have. For example:
- If triangle \(DEF\) has its vertices moved from \(ABC\), and it appears directly above/below or laterally aligned with \(ABC\), this may indicate a rotational or reflection move followed by a translation.
- Analyzing shifts either to the right, left, up, or down will provide significant insight into the nature of the transformation.
Based on typical transformations:
The correct choice is usually characterized by a transformation that maintains rigid motion properties (i.e., the shape and size of the triangle remain constant).
Assuming triangle \(ABC\) has been transformed into triangle \(DEF\) through reflection and translation (and without loss of generality, without coordinates), the answer that seems most likely plausible based on a straightforward analysis is:
**The first option:** "△ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up."
If you have specific coordinates, substituting those will give you precise confirmation. Otherwise, do these aspects guide your analysis of the transformation?