### Video Transcript

Identify the type of the following
geometric transformation.

To determine the type of
transformation mapping π΄π΅πΆπ· onto π΄ prime π΅ prime πΆ prime π· prime, weβll
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 π΄π΅πΆπ· and the image has vertices π΄ prime π΅ prime πΆ
prime π· 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 π΄π΅πΆπ· 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 π·. But we see that π· 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 π΄π΅πΆπ·, 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 π΄ maps to π΄ prime, π΅ maps to π΅ prime, πΆ maps to
πΆ prime, and π· maps to π· prime by a rotation of a set number of degrees in the
same direction.

Letβs sketch a point about which we
can rotate point π· 90 degrees counterclockwise. We label the image of π· π·
prime. The point about which we rotated
must be equidistant from π· and π· prime. Letβs apply the same 90-degree
rotation to point πΆ, as shown. After doing the same for π΄ and π΅,
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 π΄π΅πΆπ· 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.