Question

The point A (-1, 1) is rotated to become A’ (2, 2) Identify the degree of rotation in both directions for the pictured rectangle. Recall that clockwise rotations are denoted with a negative degree. (1 point)

The quadrilateral is rotated ____ counterclockwise or ____ clockwise

Answer 1

To determine the degree of rotation from point A (-1, 1) to A’ (2, 2), we can follow these steps:

1. **Calculate the original position of A and A’**:
- A is at (-1, 1).
- A’ is at (2, 2).

2. **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.

3. **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:
\[ \text{Vector A'} = (2, 2) \]

4. **Calculate angles**:
The angle from the positive x-axis to each point can be calculated using the arctan function:
\[
\theta_A = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) = 135^\circ \quad (\text{in the 2nd quadrant})
\]
\[
\theta_{A'} = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = 45^\circ \quad (\text{in the 1st quadrant})
\]

5. **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**.