Identifying whether one figure is the image of another figure under rotation can be difficult. A key aspect to observe is whether the center of rotation is the same distance from an image as it is from its preimage.
A rotation is a transformation in which a figure is turned about a fixed point P by a certain angle measure The point P is called the center of rotation. Rotations map every point A in the plane to its image such that one of the following statements is satisfied.
Remember, after performing a rotation, the preimage and the image of a point are the same distance from the center of rotation. The angle of rotation is formed by a preimage, the center of rotation, and the corresponding image.
Place the center of the protractor on P and align it with PA.
The way the protractor was placed before works when the rotation is counterclockwise. If the rotation has to be done clockwise, the protractor needs to be placed as follows.
Locate the corresponding measure on the protractor and make a small mark. In this case, the mark will be made at
Draw △ABC and its image under this rotation.
Finally, the image of △ABC under the given rotation is the triangle formed by and
With this theorem in mind, consider the following example. In the diagram, quadrilateral is the image of ABCD under a certain rotation.
Find the center and angle of rotation.
Angle of rotation: clockwise or counterclockwise.
The first step is to find the center of rotation. Remember, by definition, a point and its image under a rotation are the same distance from the center.
The center of rotation is equidistant from a point and its image.
Therefore, by the Converse of the Perpendicular Bisector Theorem, the center lies on the perpendicular bisector of for instance. Then, with the aid of a compass and a straightedge, start by constructing the perpendicular bisector of this segment.
To determine the center's exact position, draw a second segment joining a vertex and its image, for example, Then, draw the perpendicular bisector of this segment. The intersection between both perpendicular bisectors is the center of rotation.
Notice that drawing only two perpendicular bisectors is enough to find the center of rotation because all will intersect at the same point. Since the sense of rotation was not specified, both measures will be found using a protractor.
The angle of rotation is either counterclockwise or clockwise.
Recall that rotations are transformations and that transformations can be composed. Therefore, it is possible to have a composition of two or more rotations. On a geometry exercise, the following two rotations are given.
LaShay has to perform both rotations to △ABC, one after the other, but the book does not indicate the composition's order.
In real life, there are plenty of situations where rotations can be appreciated. For instance, take a look at a door.