10. Triangle \( A B C \) has the coording below. Jaime will rotate the triang rule \( (x, y) \rightarrow(y,-x) \). \[ A(-5,-1) B(-7,-1) \quad C(- \] a. Describe the possible degrees of the rotation. b. List the coordinates of \( A^{\prime} \).

Let's address each part of the problem step by step. ### **a. Possible Degrees of the Rotation** The given rotation rule is: \[ (x, y) \rightarrow (y, -x) \] This transformation can be interpreted as a rotation about the origin. To determine the angle of rotation, let's compare it with the standard rotation matrix. A standard rotation matrix for an angle \(\theta\) is: \[ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \] Applying this to a point \((x, y)\) gives: \[ (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \] Matching this to the given transformation \((y, -x)\): \[ \begin{cases} x \cos \theta - y \sin \theta = y \\ x \sin \theta + y \cos \theta = -x \end{cases} \] By equating coefficients, we find: \[ \cos \theta = 0 \quad \text{and} \quad \sin \theta = -1 \] These values correspond to: \[ \theta = 270^\circ \quad \text{(clockwise)} \quad \text{or} \quad \theta = -90^\circ \] Therefore, the possible degrees of rotation are: - **270 degrees clockwise** - **90 degrees counterclockwise** ### **b. Coordinates of \( A^{\prime} \)** Given the original coordinates of point \( A \): \[ A(-5, -1) \] Applying the rotation rule \((x, y) \rightarrow (y, -x)\): \[ A' = (-1, 5) \] So, the coordinates of \( A^{\prime} \) after the rotation are: \[ A'(-1, 5) \] --- **Summary:** - **a.** The rotation is a 270° clockwise (or equivalently, 90° counterclockwise) rotation about the origin. - **b.** The coordinates of \( A^{\prime} \) are \( (-1,\ 5) \).