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Tanya's rotation maps point [tex]\( K(24,-15) \)[/tex] to [tex]\( K^{\prime}(-15,-24) \)[/tex]. Which describes the rotation?

A. [tex]\( 90^{\circ} \)[/tex] clockwise rotation
B. [tex]\( 270^{\circ} \)[/tex] clockwise rotation
C. [tex]\( 180^{\circ} \)[/tex] rotation
D. [tex]\( 90^{\circ} \)[/tex] counterclockwise rotation

Sagot :

GinnyAnswer
To determine the type of rotation that maps point [tex]\( K(24, -15) \)[/tex] to [tex]\( K'(-15, -24) \)[/tex], we need to analyze how the coordinates change under various rotations.

1. 90 degrees clockwise rotation:
- This transforms a point [tex]\( (x, y) \)[/tex] to [tex]\( (y, -x) \)[/tex].
- Applying this to [tex]\( K(24, -15) \)[/tex]: [tex]\( (24, -15) \)[/tex] becomes [tex]\( (-15, -24) \)[/tex].

2. 270 degrees clockwise rotation:
- This is equivalent to a 90 degrees counterclockwise rotation, transforming [tex]\( (x, y) \)[/tex] to [tex]\( (-y, x) \)[/tex].
- Applying this to [tex]\( K(24, -15) \)[/tex]: [tex]\( (24, -15) \)[/tex] becomes [tex]\( (15, 24) \)[/tex].

3. 180 degrees rotation:
- This transforms a point [tex]\( (x, y) \)[/tex] to [tex]\( (-x, -y) \)[/tex].
- Applying this to [tex]\( K(24, -15) \)[/tex]: [tex]\( (24, -15) \)[/tex] becomes [tex]\( (-24, 15) \)[/tex].

4. 90 degrees counterclockwise rotation:
- This transforms a point [tex]\( (x, y) \)[/tex] to [tex]\( (-y, x) \)[/tex].
- Applying this to [tex]\( K(24, -15) \)[/tex]: [tex]\( (24, -15) \)[/tex] becomes [tex]\( (15, 24) \)[/tex].

On examining the transformations, we see that the coordinates transform [tex]\( K(24, -15) \)[/tex] to [tex]\( K'(-15, -24) \)[/tex] correctly through a 90 degrees clockwise rotation. Thus, the correct description of the rotation that maps [tex]\( K \)[/tex] to [tex]\( K' \)[/tex] is:

90 degrees clockwise rotation.
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