Question

a composition of transformations maps pre - image efgh to final image efgh. the first transformation for this composition is _______, and the second transformation is a reflection across line c.
a 180° rotation about point g
a 180° rotation about point h
a translation down and to the left
a translation up and to the right

Explanation:

Step1: Analyze rotation options

• If it were a \(180^{\circ}\) rotation about point \(G\), the position of other points relative to \(G\) would change in a specific way. But looking at the pre - image \(EFGH\) and the intermediate image (before reflection), a \(180^{\circ}\) rotation about \(G\) does not map \(EFGH\) to the correct intermediate position.
• If it were a \(180^{\circ}\) rotation about point \(H\), the orientation and position of the figure \(EFGH\) with respect to \(H\) also does not match the transformation to the intermediate image.

Step2: Analyze translation options

• A translation is a slide. If we consider the direction of the transformation from \(EFGH\) to the intermediate image (before reflection across line \(c\)):
• A translation down and to the left: By observing the relative positions of the vertices of \(EFGH\) and the intermediate image (before reflection), we can see that the pre - image \(EFGH\) is moved in a direction that is down (as per the vertical arrow - like reference in the figure) and to the left (comparing the horizontal positions of corresponding vertices) to get to the intermediate image (before reflection across line \(c\)).
• A translation up and to the right is the opposite direction of what is needed to map \(EFGH\) to the intermediate image (before reflection).

Answer:

a translation down and to the left