To determine the degree of rotation from point A (-1, 1) to A’ (2, 2), we can follow these steps:
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Calculate the original position of A and A’:
- A is at (-1, 1).
- A’ is at (2, 2).
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Find the center of rotation. Since the problem does not specify a center of rotation, we will use a reference point. The center of a rectangle formed by points may typically be the midpoint of the diagonal if the quadrilateral is a rectangle. However, for this problem, let’s consider the origin (0,0) as a common reference.
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Convert points to polar coordinates or use vector reasoning:
- The position of A in relation to the origin is a vector: \[ \text{Vector A} = (-1, 1) \]
- The position of A’ in relation to the origin is: \[ \text{Vector A'} = (2, 2) \]
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Calculate angles: The angle from the positive x-axis to each point can be calculated using the arctan function: \[ \theta_A = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) = 135^\circ \quad (\text{in the 2nd quadrant}) \] \[ \theta_{A'} = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = 45^\circ \quad (\text{in the 1st quadrant}) \]
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Find the difference in angles:
- If we consider the positive direction of rotation as counterclockwise (CCW) and negative as clockwise (CW), we can find the rotation.
- From 135° (A) to 45° (A'):
- CCW rotation would be \(-90^\circ\).
- CW rotation would be \(270^\circ\).
Thus, the final result is a rotation:
- The quadrilateral is rotated 90° counterclockwise or -270° clockwise.
In concise form:
- The quadrilateral is rotated 90° counterclockwise or -270° clockwise.
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