Answer: The option that is correct is
(C) T0,2(x, y)∘ R0, 270°.
Step-by-step explanation: We need to choose the transformation rule that defines how rectangle PQRS is transformed into P''Q''R''S''.
Observing the graph, we can identify that
the points of rectangle PQRS are P(-3, -5), Q(-2, -5), R(-2, -1) and S(-3, -1).
Conversely, the vertices of rectangle P''Q''R''S'' are P''(-5, 5), Q''(-5, 4), R''(-1, 4) and S(-1, 5).
The rectangle PQRS resides in Quadrant III while P''Q''R''S'' is located in Quadrant II,indicating that the rotation could either be 90° clockwise or 270° counterclockwise around the origin.
Among the provided choices, there is no 90° rotation available, hence we will consider the 270° counterclockwise rotation around the origin.
After executing this rotation, the points transform based on the rule
(x, y) ⇒ (y, -x).
Thus, the vertices of the resulting rectangle P'Q'R'S', after a 270° counterclockwise rotation around the origin, become
P(-3, -5) ⇒ P'(-5, 3),
Q(-2, -5) ⇒ Q'(-5, 2),
R(-2, -1) ⇒ R'(-1, 2),
S(-3, -1) ⇒ S'(-1, 3).
To align the vertices of P'Q'R'S' with those of P''Q''R''S'', we must add 2 units to the y-coordinate of all vertices , leading to
P'(-5, 3) ⇒ P''(-5, 3+2) = P''(-5, 5),
Q'(-5, 2) ⇒ Q''(-5, 2+2) = Q''(-5, 4),
R'(-1, 2) ⇒ R''(-1, 2+2) = R''(-1, 4),
S'(-1, 3) ⇒ S''(-1, 3+2) = S''(-1, 5).
Thus, the required transformation rule becomes
a rotation through 270° counterclockwise around the origin coupled with a translation of (x, y) ⇒ (x, y+2).
Since rigid transformations are documented from right to left, option (C) is indeed correct.
