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:
- Reflect across the y-axis:
- A becomes (1, 3)
- B becomes (2, -1)
- C becomes (-1, 0)
- 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:
- 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)
- 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:
- 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)
- 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:
- Rotate 180 degrees:
- A(-1, 3) becomes (1, -3)
- B(-2, -1) becomes (2, 1)
- C(1, 0) becomes (-1, 0)
- 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.