Questions: A star in the coordinate plane has two of its points at (3,-5) and (-2,-7). After a rotation, the first point moved to (-5,-3). Where did the other point move? How many degrees was the star rotated?

Solution Steps

To find the angle of rotation that maps the point \((3, -5)\) to \((-5, -3)\), we calculate the angle in radians as follows:

\[ \theta = \arctan\left(\frac{-3}{-5}\right) - \arctan\left(\frac{-5}{3}\right) = -\frac{\pi}{2} \text{ radians} \]

Converting this to degrees gives:

\[ \theta = -90^\circ \]

Next, we apply the same rotation to the second point \((-2, -7)\). The rotation matrix for an angle of \(-90^\circ\) is given by:

\[ R = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \]

Multiplying this matrix by the coordinates of the second point:

\[ \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} -2 \\ -7 \end{bmatrix} = \begin{bmatrix} -7 \\ 2 \end{bmatrix} \]

Thus, the new coordinates of the second point after rotation are \((-7, 2)\).

Final Answer