Question

what is the final transformation in the composition of transformations that maps pre - image g h j k to image g h j k?
○ a translation to the right
○ a reflection across line m
○ a 90° rotation about point g
○ a 180° rotation about point g

Explanation:

1. Analyze the pre - image \( GHJ K \) and the intermediate image (after the first transformation) and then the final image \( G'H'J'K' \).
2. For a translation to the right, the orientation of the figure would not change in terms of rotation about a point, and the relative position of the vertices with respect to point \( G \) would not match the final image.
3. A reflection across line \( m \) would create a mirror - image across line \( m \), but the final image \( G'H'J'K' \) does not look like a reflection across line \( m \) from the intermediate image.
4. A \( 90^\circ \) rotation about point \( G \) would change the orientation of the figure in a way that the sides and angles would not align with \( G'H'J'K' \).
5. A \( 180^\circ \) rotation about point \( G \) means that each point \( (x,y) \) relative to \( G \) in the pre - image (or intermediate image) will be mapped to a point \( (-x,-y) \) relative to \( G \). This rotation will flip the figure around point \( G \) such that the final image \( G'H'J'K' \) is obtained. By comparing the positions of the vertices of the intermediate image (after the first transformation) and the final image \( G'H'J'K' \) with respect to point \( G \), we can see that a \( 180^\circ \) rotation about \( G \) is the correct final transformation.

Answer:

a \( 180^\circ \) rotation about point \( G \)