To determine the rotation that maps point [tex]\( M(9, -1) \)[/tex] to [tex]\( M^{\prime}(-9, 1) \)[/tex], we can examine the effect of different rotations on point [tex]\( M \)[/tex].
### 1. Rotating by [tex]\( 180^\circ \)[/tex]
A rotation of [tex]\( 180^\circ \)[/tex] around the origin will change the sign of both coordinates. Thus, point [tex]\( (x, y) \)[/tex] would transform to [tex]\( (-x, -y) \)[/tex].
- Starting with [tex]\( M(9, -1) \)[/tex]:
[tex]\[
M \text{ after } 180^\circ \text{ rotation } = (-9, 1)
\][/tex]
This matches [tex]\( M^{\prime}(-9, 1) \)[/tex].
### 2. Rotating by [tex]\( 270^\circ \)[/tex] clockwise
A [tex]\( 270^\circ \)[/tex] clockwise rotation is equivalent to a [tex]\( 90^\circ \)[/tex] counterclockwise rotation. This will swap the coordinates and change the sign of the new y-coordinate. Thus, point [tex]\( (x, y) \)[/tex] would transform to [tex]\( (y, -x) \)[/tex].
- Starting with [tex]\( M(9, -1) \)[/tex]:
[tex]\[
M \text{ after } 270^\circ \text{ clockwise rotation } = (-1, -9)
\][/tex]
This does not match [tex]\( M^{\prime}(-9, 1) \)[/tex].
### 3. Rotating by [tex]\( 90^\circ \)[/tex] counterclockwise
A [tex]\( 90^\circ \)[/tex] counterclockwise rotation will swap the coordinates and change the sign of the new x-coordinate. Thus, point [tex]\( (x, y) \)[/tex] would transform to [tex]\( (-y, x) \)[/tex].
- Starting with [tex]\( M(9, -1) \)[/tex]:
[tex]\[
M \text{ after } 90^\circ \text{ counterclockwise rotation } = (1, 9)
\][/tex]
This does not match [tex]\( M^{\prime}(-9, 1) \)[/tex].
### 4. Rotating by [tex]\( 90^\circ \)[/tex] clockwise
A [tex]\( 90^\circ \)[/tex] clockwise rotation will swap the coordinates and change the sign of the new y-coordinate. Thus, point [tex]\( (x, y) \)[/tex] would transform to [tex]\( (y, -x) \)[/tex].
- Starting with [tex]\( M(9, -1) \)[/tex]:
[tex]\[
M \text{ after } 90^\circ \text{ clockwise rotation } = (-1, 9)
\][/tex]
This does not match [tex]\( M^{\prime}(-9, 1) \)[/tex].
Based on the examination of these rotations, we can conclude that the rotation that maps point [tex]\( M(9, -1) \)[/tex] to [tex]\( M^{\prime}(-9, 1) \)[/tex] is a [tex]\( 180^\circ \)[/tex] rotation.
So, the correct answer is:
[tex]\[ 180^\circ \text{ rotation} \][/tex]
