Question

Determine the congruence transformation used on A
(-1,3)B(-2,-1)C(1,0) that resulted in D(3,-1)E(4,3)F(1,2).

a. ABC was rotated -180 degrees ( clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up
b. ABC as shifted 2 units to the left and 2 units up, then reflected across the y-axis
c. ABC was shifted 2 units to the right and 2 units up, then rotated 180 degrees (counterclockwise)
d. ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up

Answer 1

To determine the correct congruence transformation used on triangle ABC that resulted in triangle DEF, we can follow these steps:

1. Identify the coordinates of each point:

- A = (-1, 3)
- B = (-2, -1)
- C = (1, 0)
- D = (3, -1)
- E = (4, 3)
- F = (1, 2)

2. Calculate the difference in the coordinates from ABC to DEF.

3. Check each transformation option given and apply it to see if we arrive at DEF.

### Analyzing the transformations:

**Option a**: Rotate 180 degrees, then shift 2 units right and 2 units up.

1. **Rotate ABC 180 degrees**:
- A: (-1, 3) becomes (1, -3)
- B: (-2, -1) becomes (2, 1)
- C: (1, 0) becomes (-1, 0)

2. **Shift 2 units to the right and 2 units up**:
- A': (1, -3) -> (3, -1)
- B': (2, 1) -> (4, 3)
- C': (-1, 0) -> (1, 2)

This results in points D(3, -1), E(4, 3), F(1, 2) - this matches DEF.

**Option b**: 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)

This results in points (3, 5), (4, 1), (1, 2), which does not match DEF.

**Option c**: 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)

This results in (-1, -5), (0, -1), (-3, -2), which does not match DEF.

**Option d**: Reflect across the y-axis, then shift 2 units to the right and 2 units up.

1. **Reflect ABC across the y-axis**:
- 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, 5)
- B: (2, -1) becomes (4, 1)
- C: (-1, 0) becomes (1, 2)

This results in (3, 5), (4, 1), (1, 2), which does not match DEF.

### Conclusion:

Only **Option a** yields the correct transformation resulting in D(3, -1), E(4, 3), F(1, 2). Therefore, the correct answer is:

**a. ABC was rotated -180 degrees (clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up.**