Answer: The complete statement is:
ΔABC is rotated through an angle of 180° about the origin to become ΔA'B'C'. Then, ΔA'B'C' is translated 3 units down to become ΔA''B''C''. Because the transformations are ROTATION and TRANSLATION, the pre image and image are SIMILAR.
Step-by-step explanation: We are given a diagram that shows a pre-image ΔABC and its image ΔA′′B′′C′′ after a series of transformations.
We are given to find the transformations from ΔABC to ΔA'B'C' and then from ΔA'B'C' to ΔA''B''C''.
The co-ordinates of the vertices of ΔABC are A(1, -1), B(4, -2) and C(7, 2).
The co-ordinates of the vertices of ΔA'B'C' are A'(-1, 1), B'(-4, 2) and C'(-7, -2).
The co-ordinates of the vertices of ΔA''B''C'' are A''(-1, -2), B''(-4, -1) and C''(-7, -5).
We can see that
the vertices of ΔABC follow the rule (x, y) ⇒ (-x, -y) to form the vertices of ΔA'B'C'.
So, ΔABC is rotated about the origin (0, 0) through an angle of 180° to form ΔA'B'C'.
Again, the vertices of ΔA'B'C' follow the rule (x, y) ⇒ (x, y-3) to form the vertices of ΔA''B''C''.
So, ΔA'B'C' is translated 3 units down to form ΔA''B''C''.
Thus, the required transformations are
ΔABC is rotated through an angle of 180° about the origin to become ΔA'B'C'. Then, ΔA'B'C' is translated 3 units down to become ΔA''B''C''. Because the transformations are ROTATION and TRANSLATION, the pre image and image are SIMILAR.