Question

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)

Answer 1

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.

Answer 2

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

Further explanation:

A translation is a transformation that transforms the figure with a fixed distance in the same direction.

A rotation is the transformation that rotates the figure with given angles.

Given:

It is given that the two transformations that maps pre-image PQRS to image ${\text{P''Q''R''S''}}$ .

Step by step explanation:

Step 1:

It can be seen from the given figure that that the pre image is in the third quadrant and the image is the second quadrant.

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

Therefore, the rotation is in the counter clockwise direction of $270^\circ$ .

Step 2:

The rotation of $270^\circ$  in the counter clockwise direction represents the coordinates as,  

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

It can be seen that the coordinate of the point $P\left({-3,-5}\right)$  and $P'\left({-5,5}\right)$ .

If $x$  is $-3$  and $y$  is $-5$ .

Then after rotation of $270^\circ$  counterclockwise $\left({x,y}\right)\to\left({y,-x}\right)$  as,

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

Now with the translation of 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 given transformation is the rotation of $270^\circ$  counterclockwise followed by translation by $\left({0,2}\right)$ .

Therefore, this is the composition of transformation.

The rigid transformation can be written from right to left.

Therefore, the rotation $270^\circ$  counterclockwise would be written on the right side and the translation would be written on the left side of the composition.

The composition of the given transformation can be written as,

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

Therefore, option C is correct.

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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.