We are asked which of the given sequences maps rectangle $A B C D$ to $A^{'} B^{'} C^{'} D^{'} .$

Let's look at the first sequence.

Sequence A

In this sequence, we will first translate rectangle

A B C D

6

units down.

Next, we are asked to rotate the rectangle

9 0^{\circ}

about the origin. To perform a rotation, we need to change the

x -

and

y -

coordinates of the points of rectangle

A B C D

as shown in the following table.

Angle of Rotation	Rule
$9 0^{\circ}$	$(x, y) \to (- y, x)$
$18 0^{\circ}$	$(x, y) \to (- x, - y)$
$27 0^{\circ}$	$(x, y) \to (y, - x)$

We want to rotate our rectangle

A B C D

9 0^{\circ}

about the origin. Let's do it!

\underline{(x, y)} A (2, - 8) B (4, - 8) C (4, - 9) D (2, - 9) \to \to \to \to \to \underline{(- y, x)} A (8, 2) B (8, 4) C (9, 4) D (9, 2)

We can now plot the obtained points and draw the image of the rectangle after the rotation!

The final step of Sequence A is to translate rectangle

A B C D

6

units to the right. Notice that moving the rectangle to the right will not map it to

A^{'} B^{'} C^{'} D^{'} .

Thus, Sequence A does not map $A B C D$ onto $A^{'} B^{'} C^{'} D^{'} .$ Next, we can check out Sequence B.

Sequence $B$

In Sequence B we are asked to start with a reflection of $A B C D$ across the $x -$ axis. To reflect the rectangle we need to reflect all its vertices.

To do so, we will map them to the other side of the line of reflection — the $x -$ axis. Note that the reflected points will be the same distance from the line of reflection as the original points, but will be located on its other side. Let's do it!

We can connect the reflected points in order to obtain the reflection of our rectangle.

Next, we are asked to translate the reflected rectangle

A B C D

6

units down.

Now, we will reflect the obtained rectangle across the

y -

axis.

Finally, we can translate rectangle

A B C D

6

units up.

We can see that Sequence B does not map

A B C D

A^{'} B^{'} C^{'} D^{'} .

Sequence $C$

Sequence C starts with a $9 0^{\circ}$ rotation about the origin of the rectangle $A B C D .$ Just like before, we need to change the $x -$ and $y -$ coordinates of the points of the rectangle as shown in the following table.

Angle of Rotation	Rule
$9 0^{\circ}$	$(x, y) \to (- y, x)$
$18 0^{\circ}$	$(x, y) \to (- x, - y)$
$27 0^{\circ}$	$(x, y) \to (y, - x)$

We want to rotate our rectangle

A B C D

9 0^{\circ}

about the origin. Let's do it!

\underline{(x, y)} A (2, - 2) B (4, - 2) C (4, - 3) D (2, - 3) \to \to \to \to \to \underline{(- y, x)} A (2, 2) B (2, 4) C (3, 4) D (3, 2)

We can draw the rectangle after the rotation!

Next, we are asked to reflect rectangle

A B C D

across the

x -

axis.

Finally, we will translate rectangle

A B C D

6

units to the left and

6

units up.

We found that Sequence C maps

A B C D

A^{'} B^{'} C^{'} D^{'} .

Sequence $D$

In this sequence, we will first translate rectangle

A B C D

6

units left.

Next, let's reflect the rectangle across the

y -

axis.

Now, we will translate rectangle

A B C D

6

units down.

Finally, we will rotate the rectangle

18 0^{\circ}

about the origin.

We can see that Sequence D does not map rectangle

A B C D

A^{'} B^{'} C^{'} D^{'} .

Let's summarize our results.

Sequence	Does it map $A B C D$ to $A^{'} B^{'} C^{'} D^{'}$ ?
Sequence $A$	No
Sequence $B$	No
Sequence $C$	Yes
Sequence $D$	No

We found that only Sequence C maps rectangle $A B C D$ to $A^{'} B^{'} C^{'} D^{'} .$

Exercise

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