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Congruence and Rigid Motion

Describing Rotations



A rotation is a type of transformation that rotates or turns an object around a point called the center of rotation. The number of degrees the figure is rotated, is called the angle of rotation.

Using the origin as the center of rotation, rotate the rectangle counterclockwise.

Show Solution

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

One of these points can now be rotated 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.


The triangle has been rotated to create Determine the angle and center of rotation used for the transformation.

Show Solution

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, This can be confirmed by looking at the remaining sides, that create straight angles with their transformed counterparts.

Thus, the angle of rotation is 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 Consequently, must be the center of rotation.


Rotational Symmetry

A figure is said to have rotational symmetry if it's possible to rotate it less than without changing its appearance. For example, the digit can be rotated and still look the same. An equilateral triangle can be rotated to produce an identical figure, while a square can be rotated with the same result.

Consider the parallelogram shown. Find the angle of rotation that will carry it onto itself.

Show Solution

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

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 about its center.

Because the image lies on top of the preimage, we can conclude that the desired rotation is around the center of the parallelogram.


Composite Transformations

It's possible to perform different transformations on the same object, a composite transformation. Each transformation is done separately to create the final image. The right triangle in the coordinate system it's first rotated counterclockwise, with the right angle's vertex as center of rotation, and then translated unit to the right and unit down.

It's also possible to rotate, and then reflect the image. The same triangle can be rotated clockwise around the right angle's vertex, and then be reflected in the -axis, as shown below.

For some transformations, like translations and some rotations, it doesn't matter the order in which they are performed. As a general rule, however, it's advisable to always perform the transformations in the given order to ensure that the final image is correct.
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