Question

Where would the point be located when the point -2,1 is rotated -90 degrees around the center of rotation which is point 1,5?

a) (-1, 6)

b) (6, -1)

c) (4, 1)

d) (-4, 1)

Answer 1

To find the new location of a point after a -90-degree rotation around a point other than the origin, you must first translate the point so that the center of rotation is the origin, rotate using the appropriate rotation matrix, and then translate back. The point (-2,1) rotated -90 degrees around (1,5) ends up at the point (5,2), which is not in the provided options.

Step-by-step explanation:

To determine the location of a point after a rotation around a center of rotation, we first translate the center of rotation to the origin and then apply the rotation matrix. Rotating the point (-2,1) by -90 degrees around the center (1,5) involves two steps: a translation such that the center of rotation becomes the origin, and then applying a rotation matrix to the translated point.

First, translate point (-2,1) by subtracting the center of rotation (1,5), yielding a new point (-3,-4). Next, we apply the rotation matrix for a -90 degrees rotation, which is:

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

Multiplying the matrix by the point (-3,-4) gives us the rotated point (4,-3). Finally, we reverse the original translation by adding the center of rotation (1,5), which yields the final position (5,2). None of the given options match this result.