To determine which angle corresponds to angle \( K \) in the rotation of pentagon \( ABCDE \) 90° counterclockwise around point \( F \), we can analyze the coordinates and their positions before and after the rotation.
Let's list the points of pentagon \( ABCDE \):
- \( A(0, 5) \)
- \( B(-2, 4) \)
- \( C(-1, 2) \)
- \( D(1.1, 3.1) \)
- \( E(1.4, 5.2) \)
Now, point \( F(3, 3) \) is our center of rotation. When rotating a point \( P(x, y) \) 90° counterclockwise around point \( F(a, b) \), the new coordinates \( P' \) are given by:
\[ P' = (a - (y - b), b + (x - a)) \]
We need to examine the details of the rotation and identify which point of \( ABCDE \) corresponds to point \( K(2, -2) \) of the dotted pentagon. Let's calculate the new coordinates for each point after rotating 90° counterclockwise around point \( F \).
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For point \( A(0, 5) \): \[ A' = (3 - (5 - 3), 3 + (0 - 3)) = (3 - 2, 3 - 3) = (1, 0) \]
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For point \( B(-2, 4) \): \[ B' = (3 - (4 - 3), 3 + (-2 - 3)) = (3 - 1, 3 - 5) = (2, -2) \]
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For point \( C(-1, 2) \): \[ C' = (3 - (2 - 3), 3 + (-1 - 3)) = (3 - (-1), 3 - 4) = (4, -1) \]
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For point \( D(1.1, 3.1) \): \[ D' = (3 - (3.1 - 3), 3 + (1.1 - 3)) = (3 - 0.1, 3 - 1.9) = (2.9, 1.1) \]
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For point \( E(1.4, 5.2) \): \[ E' = (3 - (5.2 - 3), 3 + (1.4 - 3)) = (3 - 2.2, 3 - 1.6) = (0.8, 1.4) \]
After the rotation, we notice that point \( B' \) has the coordinates \( (2, -2) \), which matches the coordinates of point \( K \).
Thus, angle \( K \) corresponds to angle \( B \).
So the answer is:
2