Question

identify which sequence of transformations could map pentagon abcde onto pentagon abcde, as shown below. 1) dilation followed by a rotation 2) translation followed by a rotation 3) line reflection followed by a translation 4) line reflection followed by a line reflection

Explanation:

• A dilation changes the size of a figure. Since the two pentagons appear to be the same size, option 1 (dilation followed by a rotation) is incorrect.
• A translation slides a figure without rotating or reflecting it. Just a translation followed by a rotation would not map the pentagon \(ABCDE\) to \(A''B''C''D''E''\) as the orientation change is more complex than a simple rotation after translation, so option 2 (translation followed by a rotation) is incorrect.
• A line - reflection flips a figure over a line. If we first reflect pentagon \(ABCDE\) over a horizontal line (which would flip it vertically) and then translate it (move it to the new position), we can map \(ABCDE\) to \(A''B''C''D''E''\).
• Two line - reflections (if not over parallel lines) would be equivalent to a rotation (if the lines are intersecting) or a translation (if the lines are parallel). But in this case, the transformation is best explained by a reflection (to flip the orientation) followed by a translation (to move it to the new location) rather than two reflections.

Answer:

1. line reflection followed by a translation