Question

Missy’s rotation maps point K(17, –12) to K’(12, 17). Which describes the rotation?
270 degrees counterclockwise rotation
90 degrees counterclockwise rotation
90 degrees clockwise rotation
180 degrees rotation

Answer 1

To determine the type of rotation that maps point \( K(17, -12) \) to \( K'(12, 17) \), we can analyze the coordinates before and after the rotation.

1. **Original Point \( K(17, -12) \)**:
- \( x_1 = 17 \)
- \( y_1 = -12 \)

2. **Point After Rotation \( K'(12, 17) \)**:
- \( x_2 = 12 \)
- \( y_2 = 17 \)

The transformations for rotations in the coordinate plane around the origin are as follows:
- **90 degrees counterclockwise**: \( (x, y) \rightarrow (-y, x) \)
- **90 degrees clockwise**: \( (x, y) \rightarrow (y, -x) \)
- **180 degrees rotation**: \( (x, y) \rightarrow (-x, -y) \)
- **270 degrees counterclockwise**: \( (x, y) \rightarrow (y, -x) \) which is equivalent to a 90 degrees clockwise rotation.

Now let's check which rotation applies:

### Check 90 degrees counterclockwise
Applying this rotation to \( K(17, -12) \):
\[
(-(-12), 17) \rightarrow (12, 17) \rightarrow K'
\]
This matches \( K'(12, 17) \).

### Check 90 degrees clockwise
Applying this rotation to \( K(17, -12) \):
\[
(-12, 17) \rightarrow \text{not} K'
\]

### Check 180 degrees
Applying this rotation:
\[
(-17, 12) \rightarrow \text{not} K'
\]

### Check 270 degrees counterclockwise
This is the same as 90 degrees clockwise:
Applying this rotation:
\[
(-12, 17) \rightarrow \text{not} K'
\]

Thus, the transformation that maps \( K(17, -12) \) to \( K'(12, 17) \) is a **90 degrees counterclockwise rotation**.