Describing Rotations

To rotate the rectangle, we can focus on its vertices.

One of these points can now be rotated $90^{\circ }$ counterclockwise around the origin.

Similarly, we can rotate the other points.

Lastly, we can draw the image, using the blue points as the vertices of the rectangle.

Let's start by finding the angle of rotation. One clue we have is that the corresponding short sides are both vertical. Thus, the triangle must have rotated either a half- or a full-turn. Since a full rotation would map the original triangle onto itself, it must have been a half rotation, $180^{\circ }.$ This can be confirmed by looking at the remaining sides, that create straight angles with their transformed counterparts.

Thus, the angle of rotation is $180^{\circ }.$ Next, we can address the center of rotation. In a rotation, an entire figure rotates about the center of rotation. Thus, everything moves except the center of rotation itself. Here, we can identify the point that remained in the same location after the transformation.

This point is the vertex $(3,2).$ Consequently, $(3,2)$ must be the center of rotation.

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel and congruent. Additionally, opposite angles are congruent. To begin, let's try to determine the center of rotation. Suppose it is the upper right corner. Let's rotate it clockwise $180^{\circ }.$

Notice that, although the image is a duplicate of itself, it's in a different location. That means this rotation did not map it onto itself. The same is true if we use any other vertex as the center of rotation. Instead, let's try rotating it clockwise $180^{\circ }$ about its center.

Because the image lies on top of the preimage, we can conclude that the desired rotation is $180^{\circ }$ around the center of the parallelogram.