To determine which rotation maps point [tex]\( K(8, -6) \)[/tex] to [tex]\( K'(-6, -8) \)[/tex], let's consider the transformations corresponding to each type of rotation:
1. 180° counterclockwise rotation or 180° clockwise rotation:
- For these rotations, the transformation is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
- Applying this to point [tex]\( K(8, -6) \)[/tex]:
[tex]\[
(8, -6) \rightarrow (-8, 6)
\][/tex]
- Point [tex]\( (-8, 6) \)[/tex] does not match [tex]\( K'(-6, -8) \)[/tex], so it is not a 180° rotation.
2. 90° clockwise rotation:
- For a 90° clockwise rotation, the transformation is [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
- Applying this to point [tex]\( K(8, -6) \)[/tex]:
[tex]\[
(8, -6) \rightarrow (-6, -8)
\][/tex]
- Point [tex]\( (-6, -8) \)[/tex] matches [tex]\( K'(-6, -8) \)[/tex], so it fits this transformation.
3. 90° counterclockwise rotation:
- For a 90° counterclockwise rotation, the transformation is [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
- Applying this to point [tex]\( K(8, -6) \)[/tex]:
[tex]\[
(8, -6) \rightarrow (6, 8)
\][/tex]
- Point [tex]\( (6, 8) \)[/tex] does not match [tex]\( K'(-6, -8) \)[/tex], so it is not a 90° counterclockwise rotation.
Based on the transformations, the rotation that correctly maps point [tex]\( K(8, -6) \)[/tex] to point [tex]\( K'(-6, -8) \)[/tex] is the 90° clockwise rotation.
