Answer: The correct option is
(C) T0,2(x, y)∘ R0, 270°.
Step-by-step explanation: We are given to select the rule that describes the composition of transformations that maps rectangle PQRS to P''Q''R''S''.
From the graph, we note that
the co-ordinates of the vertices of rectangle PQRS are P(-3, -5), Q(-2, -5), R(-2, -1) and S(-3, -1).
and the co-ordinates of the vertices of rectangle P''Q''R''S'' are P''(-5, 5), Q''(-5, 4), R''(-1, 4) and S(-1, 5).
Since the rectangle PQRS lies in the Quadrant III and P''Q''R''S'' lies in Quadrant II, so the rotation will be either 90° clockwise or 270° counterclockwise about the origin.
In the given options, we do not have rotation of 90°, so will consider the rotation through 270° counterclockwise about the origin.
Now, after this rotation, the vertices will transform according to the rule
(x, y) ⇒ (y, -x).
Therefore, the vertices of the image rectangle P'Q'R'S', after rotation through 270° counterclockwise about the origin, becomes
P(-3, -5) ⇒ P'(-5, 3),
Q(-2, -5) ⇒ Q'(-5, 2),
R(-2, -1) ⇒ R'(-1, 2),
S(-3, -1) ⇒ S'(-1, 3).
Now, to make the vertices of P'Q'R'S' coincide with the vertices of P''Q''R"S", we need to add 2 units to the y co-ordinate of each vertex , so that
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 is
rotation through 270° counterclockwise about the origin and a translation of (x, y) ⇒ (x, y+2).
Since the rigid transformations are written from right to left, so option (C) is correct.
