Question Video: Identifying a Given Transformation Mathematics

Video Transcript

Identify the type of the following geometric transformation.

To determine the type of transformation mapping 𝐴𝐵𝐶𝐷 onto 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime, we’ll consider which of the properties of both the object and the image have remained the same and which have been changed.

First, we can see that the vertices of the object occur in the same order as the vertices in the image. In other words, the first quadrilateral has vertices 𝐴𝐵𝐶𝐷 and the image has vertices 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime in the same order counterclockwise. This means that it cannot be a reflection. Otherwise, the vertices would have changed order, as shown in the following sketch where we see 𝐴𝐵𝐶𝐷 reflected in a mirror line.

Second, we can see that the vertices of the object do not occur in the same relative positions to one another as the vertices in the image. For example, the vertex pointing down in the object is 𝐷. But we see that 𝐷 prime is in the top-right position in the image. This means that it cannot be a translation. Otherwise, the vertices of the image would be oriented in the same direction as the vertices of 𝐴𝐵𝐶𝐷, as shown in the following sketch of a translation.

The only type of transformation we have not yet considered is a rotation. A rotation is when an object is moved around a fixed point by a set number of degrees, either clockwise or counterclockwise, to give a new image. Thus, a rotation will change the orientation of the vertices. So, if this is a rotation, there must exist a point about which 𝐴 maps to 𝐴 prime, 𝐵 maps to 𝐵 prime, 𝐶 maps to 𝐶 prime, and 𝐷 maps to 𝐷 prime by a rotation of a set number of degrees in the same direction.

Let’s sketch a point about which we can rotate point 𝐷 90 degrees counterclockwise. We label the image of 𝐷 𝐷 prime. The point about which we rotated must be equidistant from 𝐷 and 𝐷 prime. Let’s apply the same 90-degree rotation to point 𝐶, as shown. After doing the same for 𝐴 and 𝐵, we end up with a sketch like this. Finally, by joining up the vertices with edges, we get a new image that is oriented in a very similar way to the image given in the question.

Through our reasoning, we have shown why this transformation cannot be a reflection or a translation. Then, we showed how 𝐴𝐵𝐶𝐷 could be mapped onto its image by rotating a set number of degrees about a point. We note that all geometric transformations should preserve the dimensions of the object. We conclude that the transformation that best represents the given figure is a rotation.