Answer:
To show that triangle \( \triangle ABC \) is similar to triangle \( \triangle A''B''C'' \), we need to perform a series of transformations that preserve the shape of the triangles.
Given the coordinates of the vertices:
\( \triangle ABC \) has points \( (-9, 3) \), \( (-9, 6) \), and \( (0, 3) \).
\( \triangle A''B''C'' \) has points \( (3, -1) \), \( (3, -2) \), and \( (0, -1) \).
We can see that triangle \( \triangle ABC \) has been rotated 90 degrees clockwise and then translated 12 units to the right and 3 units down to obtain triangle \( \triangle A''B''C'' \). To show similarity, we need to reverse these transformations.
Thus, the correct sequence of transformations is:
1. A 180° rotation about the origin (to reverse the 90° clockwise rotation).
2. Then, a dilation by a scale factor of One-third (to reverse the translation).
So, the correct choice is: a 180° rotation about the origin, then a dilation by a scale factor of One-third.
