What two rigid transformations were performed on figure ABCD to produce the congruent figure WXYZ?

ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise).

ABCD was first rotated -270° (clockwise), then shifted 3 units to the left and 5 units up.

ABCD was first shifted 3 units to the left and 5 units up, then rotated -90° (clockwise).

ABCD was first rotated 90° (counterclockwise), then shifted 3 units to the left and 5 units up.

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

To determine the sequence of transformations that were applied to figure ABCD to produce the congruent figure WXYZ, let's analyze the transformations provided:

1. **Translation (Shift)**:
- Shifting 3 units to the left means subtracting 3 from the x-coordinates of all points.
- Shifting 5 units up means adding 5 to the y-coordinates of all points.

2. **Rotation**:
- A 90° counterclockwise rotation around the origin can be described as changing the coordinates (x, y) to (-y, x).
- A -90° (or 270° counterclockwise) rotation can be described as changing (x, y) to (y, -x).

Now let's analyze the provided sequences:

1. **ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise)**:
- **Shift**: (x, y) → (x - 3, y + 5)
- **Rotate 90° CCW**: (x - 3, y + 5) → (-(y + 5), x - 3)

2. **ABCD was first rotated -270° (clockwise), then shifted 3 units to the left and 5 units up**:
- **Rotate -270° CW (which is equivalent to 90° CCW)**: (x, y) → (-y, x)
- **Shift**: (-y, x) → (-y - 3, x + 5)

3. **ABCD was first shifted 3 units to the left and 5 units up, then rotated -90° (clockwise)**:
- **Shift**: (x, y) → (x - 3, y + 5)
- **Rotate -90° CW**: (x - 3, y + 5) → (y + 5, -(x - 3))

4. **ABCD was first rotated 90° (counterclockwise), then shifted 3 units to the left and 5 units up**:
- **Rotate 90° CCW**: (x, y) → (-y, x)
- **Shift**: (-y, x) → (-y - 3, x + 5)

To find the congruence and the right sequence of transformations that would leave the two figures congruent, it helps to consider the nature of the transformations.

The first and third transformations start with the translation followed by a rotation. The second and fourth transformations start with a rotation followed by a translation.

Given the context of the question, it is most likely that the definition aligns more with classic congruence processes, and the **first transformation is likely correct** because it follows the conventional translation followed by a rotation.

So the answer to the question is:

**ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise).**