To find the measurement of angle \( A \) after a 90° clockwise rotation of triangle \( \Delta ABC \) around point \( D(0, 2) \), we can follow these steps:
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Determine the coordinates of points A, B, and C:
- \( A(-3, 5) \)
- \( B(-5, 1) \)
- \( C(-1, 1) \)
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Determine the translation of point \( D \):
- The point of rotation \( D \) has coordinates (0, 2).
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Perform the 90° clockwise rotation: To rotate a point \( (x, y) \) around another point \( (h, k) \) by 90° clockwise, the new coordinates \( (x', y') \) can be calculated as follows: \[ x' = h + (y - k) \] \[ y' = k - (x - h) \]
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Apply the rotation to each point:
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Rotate point \( A(-3, 5) \): \[ x' = 0 + (5 - 2) = 3 \] \[ y' = 2 - (-3 - 0) = 2 + 3 = 5 \] Thus, \( A' = (3, 5) \).
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Rotate point \( B(-5, 1) \): \[ x' = 0 + (1 - 2) = -1 \] \[ y' = 2 - (-5 - 0) = 2 + 5 = 7 \] Thus, \( B' = (-1, 7) \).
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Rotate point \( C(-1, 1) \): \[ x' = 0 + (1 - 2) = -1 \] \[ y' = 2 - (-1 - 0) = 2 + 1 = 3 \] Thus, \( C' = (-1, 3) \).
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Find angle \( A' \) in triangle \( \Delta A'B'C' \): To find the angle \( A' \), we can use the properties of triangles and the fact that a triangle's angles will remain the same after a rotation.
The angle \( A \) in triangle \( \Delta ABC \) is \( 63.4° \). Since the rotation does not change the angle measures, and the shape of the triangle remains the same, angle \( A' \) will also be: \[ \text{Angle } A' = \text{Angle } A = 63.4° \]
Therefore, the measurement of angle \( A \) in the rotation of \( \Delta ABC \) \( 90° \) clockwise around point \( D \) is \( \boxed{63.4°} \).
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