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Use the image to answer the 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 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 rotated −180° (clockwise) or 180° (counterclockwise), then shifted 2 units to the right and 2 units up.

1 answer

To determine the congruence transformation used on triangle \( \triangle ABC \) that resulted in triangle \( \triangle DEF \), we must analyze the positions of the vertices of both triangles. Since I can't see the image, you should follow these steps to decide which transformation is correct based on their properties:

  1. Check Vertices Positions:

    • Identify the coordinates of points \( A, B, C \) and \( D, E, F \).

  2. Transformation Properties:

    • Reflection across the y-axis flips the x-coordinates of the points but keeps the y-coordinates the same: \((x, y) \to (-x, y)\).
    • Shifting means adding/subtracting to the x and/or y coordinates: \((x, y) \to (x + h, y + k)\).
    • Rotation by 180° (clockwise or counterclockwise) changes both x and y coordinates: \((x, y) \to (-x, -y)\).

  3. Analyze Each Option:

    • Determine if \( \triangle DEF \) appears to be a reflection and then shifted, simply shifted, or rotated about a point.
    • Compare the transformations listed against the movements of \( A, B, C \) to \( D, E, F \).

  4. Conclusion:

    • Identify which transformation aligns with the changes observed in the coordinates you recorded.

Without the specific coordinates, I can suggest:

  • If the triangles appear to be in opposite quadrants or reflected over the x-axis/y-axis, then reflection might be involved.
  • If both triangles are simply moved while maintaining the same orientation, it indicates a shift.
  • If they appear flipped and have opposite orientations, a rotation along with a shift might be the correct transformation.

After corresponding the triangle vertices and movements, select the correct option among the provided responses.