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Determine the congruence transformation used on triangle ABC that resulted in triangle DEF.

A. triangle ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up
B. triangle ABC was shifted 2 units to the right and 2 units up, then rotated 180 degrees (counterclockwise)
C. triangle ABC was shifted 2 units to the left and 2 units up, then reflected across the y-axis.
D. triangle ABC was rotated -180 degrees (clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up.

Triangle ABC
Point A: (-1, 3)
Point B: (-2, -1)
Point C: (1, 0)

Triangle DEF
Point D: (3, -1)
Point E: (4, 3)
Point F: (1, 2)

1 answer

To determine the congruence transformation used on triangle ABC to result in triangle DEF, we first need to analyze the coordinates of the vertices of both triangles.

Coordinates of triangle ABC:

  • A(-1, 3)
  • B(-2, -1)
  • C(1, 0)

Coordinates of triangle DEF:

  • D(3, -1)
  • E(4, 3)
  • F(1, 2)

Let's apply each of the possible transformations listed in the options to triangle ABC and see if we can achieve the coordinates of triangle DEF.

Option A

Reflection across the y-axis, then shift 2 units right and 2 units up:

  1. Reflect across the y-axis:
    • A becomes (1, 3)
    • B becomes (2, -1)
    • C becomes (-1, 0)
  2. Shift 2 units right and 2 units up:
    • A (1, 3) → (3, 5)
    • B (2, -1) → (4, 1)
    • C (-1, 0) → (1, 2)

After these transformations, the new coordinates of triangle A'B'C' would be:

  • A': (3, 5)
  • B': (4, 1)
  • C': (1, 2)

This does not match triangle DEF's coordinates, so Option A does not work.

Option B

Shift 2 units right and 2 units up, then rotate 180 degrees:

  1. Shift 2 units right and 2 units up:
    • A(-1, 3) becomes (1, 5)
    • B(-2, -1) becomes (0, 1)
    • C(1, 0) becomes (3, 2)
  2. Rotate 180 degrees:
    • A (1, 5) becomes (-1, -5)
    • B (0, 1) becomes (0, -1)
    • C (3, 2) becomes (-3, -2)

The new coordinates do not match triangle DEF.

Option C

Shift 2 units left and 2 units up, then reflect across the y-axis:

  1. Shift 2 units left and 2 units up:
    • A(-1, 3) becomes (-3, 5)
    • B(-2, -1) becomes (-4, 1)
    • C(1, 0) becomes (-1, 2)
  2. Reflect across the y-axis:
    • A (-3, 5) becomes (3, 5)
    • B (-4, 1) becomes (4, 1)
    • C (-1, 2) becomes (1, 2)

The new coordinates do not match triangle DEF.

Option D

Rotate -180 degrees (clockwise) or 180 degrees (counterclockwise), then shift 2 units right and 2 units up:

  1. Rotate 180 degrees:
    • A(-1, 3) becomes (1, -3)
    • B(-2, -1) becomes (2, 1)
    • C(1, 0) becomes (-1, 0)
  2. Shift 2 units right and 2 units up:
    • A (1, -3) becomes (3, -1)
    • B (2, 1) becomes (4, 3)
    • C (-1, 0) becomes (1, 2)

After this transformation, the new coordinates would be:

  • A': (3, -1)
  • B': (4, 3)
  • C': (1, 2)

This perfectly matches the coordinates of triangle DEF.

Conclusion

The correct option that describes the transformation that was applied to triangle ABC to produce triangle DEF is: D. triangle ABC was rotated -180 degrees (clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up.