To determine the direction and angle of rotation from polygon ABCD to polygon A'B'C'D', we need to analyze the coordinates of points A and A'.
Point A: (-8, 12)
Point A': (-12, -8)
Step 1: Analyze the Transformation
- The original point A is located in quadrant 2, where x is negative and y is positive.
- The transformed point A' is in quadrant 3, where both x and y are negative.
Step 2: Determine the Rotation
We can find how the coordinates change when rotated around the origin:
- A 90° clockwise rotation would transform the point (x, y) into (y, -x).
- A 90° counterclockwise rotation would transform the point (x, y) into (-y, x).
- A 180° rotation would transform the point (x, y) into (-x, -y).
- A 270° counterclockwise rotation is equivalent to a 90° clockwise rotation, as it effectively rotates the point back by 90°.
Step 3: Calculate the Transformations
Let's see if A = (-8, 12) can yield (-12, -8) through any of these transformations:
-
90° Clockwise Transformation:
\[ (x, y) \to (y, -x) \Rightarrow (-8, 12) \to (12, 8) \]
This does not yield A'.
-
90° Counterclockwise Transformation:
\[ (x, y) \to (-y, x) \Rightarrow (-8, 12) \to (-12, -8) \]
This transformation gives us A', which matches.
-
180° Counterclockwise Transformation:
\[ (x, y) \to (-x, -y) \Rightarrow (-8, 12) \to (8, -12) \]
This does not yield A'.
-
270° Counterclockwise Transformation:
A 270° counterclockwise is the same as a 90° clockwise. Therefore, this does not apply.
Conclusion
The transformation from point A to point A' is a 90° counterclockwise rotation.