Use the image to answer the question.

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 shifted 2 units to the left and 2 units up, then reflected across the y-axis.
triangle upper A upper B upper C was shifted 2 units to the left and 2 units up, then reflected across the y -axis.

△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.

Answer 1

To determine the congruence transformation used on triangle \( \triangle ABC \) to result in triangle \( \triangle DEF \), we need to analyze the possible transformations described in the response options.

1. Shifts and reflections can change the position and orientation of a triangle on a coordinate plane.
2. Rotations can alter the orientation of the triangle without changing its position.
3. For transformations to preserve congruence, any combination of translation (shift), reflection, and rotation is valid.

Now, let's consider the options:

• Option 1: Shifting 2 units left and 2 units up then reflecting across the y-axis.
• Option 2: Reflecting across the y-axis, then shifting 2 units to the right and 2 units up.
• Option 3: Shifting 2 units to the right and 2 units up, then rotating 180° (counterclockwise).
• Option 4: Rotating -180° (clockwise) or 180° (counterclockwise), then shifting 2 units to the right and 2 units up.

Without the visual information provided by the image, I can suggest that you match the positions of \( \triangle ABC \) and \( \triangle DEF \) appropriately based on the transformations listed.

For instance:

• If \( \triangle DEF \) appears to be shifted from \( \triangle ABC \) by a certain direction and amount, that may indicate options 1 or 3.
• If \( \triangle DEF \) looks like it has been turned or flipped, options involving rotation or reflection will be more relevant.

Given the combinations, reflect on the changes in position and orientation as you inspect the triangles.

Without the diagram to analyze, please carefully apply the transformations step-by-step to see which one matches the observed change from \( \triangle ABC \) to \( \triangle DEF \).

Answer 2

which one?

Answer 3

Since I can't see the image, I'll guide you through the reasoning process to identify the correct transformation based on the transformations described.

1. Identify the Transformation: Look at the positions of points \( A \), \( B \), and \( C \) compared to \( D \), \( E \), and \( F \). Determine if the triangle appears to:

   • Move up, down, left, or right (translation).
   • Flip over the y-axis or another line (reflection).
   • Turn clockwise or counterclockwise (rotation).

2. Match Movements:

   • If points in \( \triangle DEF \) appear 2 units up and some units left from \( \triangle ABC \), this suggests a shift and possibly a reflection.
   • If \( \triangle DEF \) appears flipped compared to \( \triangle ABC \), look for reflections in your options.

3. Evaluate Options:

   • Option 1: Shifting 2 units left and 2 units up, then reflecting across the y-axis.
   • Option 2: Reflecting across the y-axis first, then shifting right and up.
   • Option 3: Shifting right and up, then rotating.
   • Option 4: Rotating first, then shifting.

After performing this analysis:

• If you identify that \( \triangle DEF \) is in a position consistent with a reflection over the y-axis followed by a shift, Option 2 is correct.
• If it looks like a shift followed by a reflection and an upward move, check Option 1.

Based on common transformations:

• If the triangles appear identical in shape and size but only shifted, it may be a translation.
• If the triangles are mirror images, a reflection is involved.

Again, without the visuals, I can guide you but can't definitively choose one. If you can provide more details or observations of the triangles' positions, I can help narrow it down further. Otherwise, use the analysis to match upon inspection.