Question

One vertex of a polygon is located at (3, –2). After a rotation, the vertex is located at (2, 3).

Which transformations could have taken place? Select two options.

R0, 90°
R0, 180°
R0, 270°
R0, –90°
R0, –270°

Answer 1

To determine which rotations could have transformed the vertex from (3, –2) to (2, 3), we can apply the rotation transformations to the original vertex.

1. **Rotation \( R_{0, 90^\circ} \)**:
\[
x' = -y, \quad y' = x
\]
Applying this to (3, –2):
\[
x' = -(-2) = 2, \quad y' = 3
\]
Resulting point: (2, 3) - **This is a valid transformation.**

2. **Rotation \( R_{0, 180^\circ} \)**:
\[
x' = -x, \quad y' = -y
\]
Applying this to (3, –2):
\[
x' = -3 = -3, \quad y' = -(-2) = 2
\]
Resulting point: (-3, 2) - **This is not a valid transformation.**

3. **Rotation \( R_{0, 270^\circ} \)**:
\[
x' = y, \quad y' = -x
\]
Applying this to (3, –2):
\[
x' = -2, \quad y' = -3
\]
Resulting point: (-2, -3) - **This is not a valid transformation.**

4. **Rotation \( R_{0, -90^\circ} \)**:
\[
x' = y, \quad y' = -x
\]
Applying this to (3, –2):
\[
x' = -2, \quad y' = -3
\]
Resulting point: (-2, -3) - **This is not a valid transformation.**

5. **Rotation \( R_{0, -270^\circ} \)** (equivalent to \( R_{0, 90^\circ} \)):
\[
x' = -y, \quad y' = x
\]
Applying this to (3, –2):
\[
x' = -(-2) = 2, \quad y' = 3
\]
Resulting point: (2, 3) - **This is a valid transformation.**

Based on this analysis, the transformations that could have taken place are:
- \( R_{0, 90^\circ} \)
- \( R_{0, -270^\circ} \)

So, the correct options are:
- **R0, 90°**
- **R0, –270°**