Figure 2 is a transformation of Figure 1, and vertex X is on line n.

Question

Figure 1 and figure 2 are irregular pentagons shown side by side. Figure 1 is smaller than Figure 2 and is oriented in a direct that is opposite of Figure 2. A dotted line labeled line n is shown between Figure 1 and Figure 2, with vertex X being a shared vertex of both figures.

Which sequence of transformations could Figure 1 have undergone to result in Figure 2?

A.
a rotation of
counterclockwise about vertex X followed by a dilation through vertex X by scale factor less than 1

B.
a reflection over line n followed by a dilation through vertex X by scale factor greater than 1

C.
a rotation of
counterclockwise about vertex X followed by a dilation through vertex X by scale factor greater than 1

D.
a reflection over line n followed by a dilation through vertex X by scale factor less than 1

Answer 1

To determine the correct sequence of transformations that could have taken Figure 1 to Figure 2, let's analyze the given information:

1. Figure 1 is smaller than Figure 2. This indicates that a dilation (scaling) was involved, specifically by a scale factor greater than 1 if we were enlarging or less than 1 if we were shrinking.

2. Figure 1 is oriented in a direction opposite of Figure 2. This suggests that a reflection could have been applied, which would flip the orientation of the figure across a line.

3. Vertex X is on line n and is a shared vertex of both figures. This reinforces the idea that the reflection might be over line n, with both figures sharing that vertex.

Considering these points, we can filter the options:

A. A rotation counterclockwise about vertex X (would change orientation) followed by a dilation through vertex X by scale factor less than 1 (would make Figure 1 smaller). This option does not seem correct since it would not account for the orientation change properly.

B. A reflection over line n (this could align the orientation correctly) followed by a dilation through vertex X by scale factor greater than 1. This seems possible because the reflection could help change the orientation to match Figure 2, and then dilation could enlarge it.

C. A rotation counterclockwise about vertex X followed by a dilation through vertex X by scale factor greater than 1. Rotation would not keep the orientation changes consistent with the description provided.

D. A reflection over line n followed by a dilation through vertex X by a scale factor less than 1. This option is incorrect since it would make Figure 1 smaller, but we need to make it larger to match Figure 2.

Thus, considering all the transformations and the requirements: The correct answer is B: a reflection over line n followed by a dilation through vertex X by scale factor greater than 1.