Which rule describes the composition of transformations that maps rectangle PQRS to P''Q''R''S''? R0,270° ∘ T0,2(x, y) R0,180° ∘

Question

mash

7 days ago

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Which rule describes the composition of transformations that maps rectangle PQRS to P''Q''R''S''? R0,270° ∘ T0,2(x, y) R0,180° ∘

T2,0 (x, y) T0,2 ∘ R0,270°(x, y) R0,2 ∘ T0,180°(x, y)

Mathematics

2 answers:

Zina [12.3K] · Answer 1 · 2026-04-05T10:27:57+03:00

The correct choice is Option C $\boxed{{T_{\left({0,2}\right)}}o{R_{0,270^\circ}}\left({x,y}\right)}$ .

Additional explanation:

A translation refers to a transformation that shifts a figure a specified distance in a uniform direction.

A rotation denotes the transformation that turns the figure at given angles.

Provided:

It has been established that two transformations map the pre-image PQRS to the resulting image ${\text{P''Q''R''S''}}$ .

Stepwise explanation:

Step 1:

Examining the depicted figure, it reveals that the pre-image is situated in the third quadrant while the image now exists in the second quadrant.

The coordinates in the second quadrant include $\left({-x,y}\right)$ and in the third quadrant include $\left({-x,-y}\right)$ when $x,y$ are positive.

Consequently, the rotation occurs in the counterclockwise orientation of $270^\circ$ .

Step 2:

The counterclockwise rotation of $270^\circ$ depicts the coordinates as,

$\left({x,y}\right)\to\left({y, - x} \right)$

Observing the coordinates of points $P\left({-3,-5}\right)$ and $P'\left({-5,5}\right)$ .

If $x$ is $-3$ and $y$ is $-5$ .

Hence after executing the counterclockwise rotation $270^\circ$ indicates as, $\left({x,y}\right)\to\left({y,-x}\right)$

$\left({-3,-5}\right)\to\left({-5,3}\right)$

Following this, translating the above point to $\left({0,2}\right)$ .

$\begin{gathered}\left({-5,3}\right)\to\left({-5+0,3+2}\right)\hfill\\\left({-5,3}\right)\to\left({-5,5}\right)\hfill\\\end{gathered}$

Therefore, the transformation described is the rotation of $270^\circ$ in a counterclockwise manner followed by a translation by $\left({0,2}\right)$ .

Therefore, the overall composition of transformations.

The rigid transformations are represented from right to left.

Thus, the counterclockwise rotation $270^\circ$ should be placed on the right side while the translation appears on the left side of the composition.

The overall composition of the provided transformations can be expressed as,

${T_{\left({0,2}\right)}}o{R_{0,270^\circ}}\left({x,y}\right)$

Consequently, option C is the right choice.

For further details:

Find out more about the transformation of functions Understand the last transformation in the series of transformations mapping pre-image abcd to image a"b'c"d"? Movement downwards and to the right, upwards and to the right, a 270° rotation about point b', or a 180° rotation about point b'

Discover more regarding the midpoint of a segment

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Transformations

Keywords: transformations, dilation, translation, rotation, counterclockwise, angle, clockwise, coordinates, mapping, rigid transformation, right side, left side, quadrant, composition.

AnnZ [12.3K] · Answer 2 · 2026-04-05T10:27:17+03:00

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.