To determine the direction and angle of rotation from polygon ABCD to polygon A'B'C'D', we can analyze the coordinates of the points provided.
- Point A is at (-7, 5) and point A' is at (5, 7).
For a rotation, we typically use the origin (0, 0) as the center of rotation.
Let's consider a 90° or 270° rotation:
-
90° counterclockwise rotation:
- The formula for a point (x, y) undergoing a 90° counterclockwise rotation is (-y, x).
- Applying this to point A: \[ A(-7, 5) \rightarrow A'(y, -x) = (-5, -7) \]
- This does not match A' (5, 7).
-
90° clockwise rotation:
- Using the formula for a 90° clockwise rotation, which is (y, -x): \[ A(-7, 5) \rightarrow A'(y, -x) = (5, 7) \]
- This does match A' (5, 7).
Thus, the rotation that transforms A to A' is a 90° clockwise rotation.
Now let's consider the remaining options mentioned in your list:
-
180° rotation: This would swap the coordinates and change their signs: \[ A(-7, 5) \rightarrow A' (7, -5) \] This does not match.
-
270° clockwise rotation: This is equivalent to 90° counterclockwise, which we already checked and found does not match.
-
270° counterclockwise rotation: This would be equivalent to a 90° clockwise rotation, which we found does match.
Final determination:
- The correct answer is 90° clockwise rotation.