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 10 to 8 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 1.
△ABC
was transformed into △DEF
. Determine which statement about this transformation is correct.

Statement #1: △ABC≅△DEF
because △ABC
was shifted 2 units to the left and 3 units down, then reflected across the y-axis.

Statement #2: △ABC≅△DEF
because △ABC
was shifted 2 units to the right and 3 units down, then rotated 180°
(counterclockwise).

Statement #3: △ABC≅△DEF
because △ABC
was shifted 2 units to the right and 3 units down, then rotated −270°
(clockwise).

Answer 1

To determine which statement about the transformation from triangle \(ABC\) to triangle \(DEF\) is correct, we need to analyze each statement carefully.

1. **Statement #1**: This statement claims that triangle \(ABC\) was shifted 2 units to the left and 3 units down, then reflected across the y-axis.
- A shift to the left would subtract from the x-coordinates. If the points of \(ABC\) were shifted left and down, and then reflected across the y-axis, you can check if \(DEF\) has the same shape and size as \(ABC\). You would need the specific coordinates of triangles \(ABC\) and \(DEF\) to confirm this.

2. **Statement #2**: Here, it states that triangle \(ABC\) was shifted 2 units to the right and 3 units down, then rotated 180° counterclockwise.
- A shift to the right adds to the x-coordinates, and after shifting, a 180° rotation would indeed result in a triangle that is congruent to the original triangle \(ABC\). Again, you would need to verify with the specific coordinates to see if this change leads to the coordinates of \(DEF\).

3. **Statement #3**: This statement indicates that triangle \(ABC\) was shifted 2 units to the right and 3 units down, then rotated -270° (which is the same as a 90° clockwise rotation).
- Similar to statement #2, this transformation involves a shift to the right, which is consistent, but the subsequent rotation would also result in a triangle congruent to \(ABC\). However, the specific transformation consistency would again need checking against the actual coordinates.

To determine the correct statement, you will need to analyze the coordinates of triangles \(ABC\) and \(DEF\). If the transformations (shifts and rotations) lead to congruence in triangle shapes and sizes, then the respective statement is correct. If the changes do not hold true when you apply them to the coordinates, then that statement would be incorrect.

In conclusion, without specific coordinate values, I cannot definitively determine which statement is correct. However, if you have the coordinates of the two triangles, you can apply the transformations described in each statement and verify which one results in triangle \(DEF\).