Question

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)

Answer 1

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.**