To find the coordinates of point A' after a counterclockwise rotation of 270° about the origin, we can use the rotation formulas related to counterclockwise transformations.
The general formula for a counterclockwise rotation of θ degrees about the origin is:
- \( x' = x \cos(\theta) - y \sin(\theta) \)
- \( y' = x \sin(\theta) + y \cos(\theta) \)
For a 270° rotation, we have:
- \( \cos(270°) = 0 \)
- \( \sin(270°) = -1 \)
Thus, the formulas simplify to:
- \( x' = y \)
- \( y' = -x \)
Now, we will apply this to each of the given points to find the new coordinates after the rotation.
-
For (2, -1):
- \( x' = -1 \)
- \( y' = -2 \)
- Result: (-1, -2)
-
For (1, 1):
- \( x' = 1 \)
- \( y' = -1 \)
- Result: (1, -1)
-
For (1, 2):
- \( x' = 2 \)
- \( y' = -1 \)
- Result: (2, -1)
-
For (-1, -2):
- \( x' = -2 \)
- \( y' = 1 \)
- Result: (-2, 1)
After checking all the calculations from the given points, it’s clear that the point (2,-1) rotates to (-1, -2) (the other points do not match).
So, the new coordinates A' after a counterclockwise rotation of 270° about the origin for point \( (2, -1) \) is:
A' = (-1, -2).
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