Exercise 12 Page 320

Practice makes perfect

We are asked which of the given sequences maps rectangle ABCD to A'B'C'D'.

Let's look at the first sequence.

Sequence A

In this sequence, we will first translate rectangle ABCD 6 units down.

Next, we are asked to rotate the rectangle 90^(∘) about the origin. To perform a rotation, we need to change the x- and y-coordinates of the points of rectangle ABCD as shown in the following table.

Angle of Rotation	Rule
90^(∘)	(x,y)→ (- y,x)
180^(∘)	(x,y)→ (- x,- y)
270 ^(∘)	(x,y)→ (y,- x)

We want to rotate our rectangle ABCD 90^(∘) about the origin. Let's do it! ccc (x,y) & → & (- y, x) [0.5em] A(2,- 8) & → & A(8,2) [0.5em] B(4,- 8) & → & B(8,4) [0.5em] C(4,- 9) & → & C(9 ,4) [0.5em] D(2,- 9) & → & 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 ABCD 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 ABCD 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 ABCD 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 ABCD 6 units down.

Now, we will reflect the obtained rectangle across the y-axis.

Finally, we can translate rectangle ABCD 6 units up.

We can see that Sequence B does not map ABCD to A'B'C'D'.

Sequence C

Sequence C starts with a 90^(∘) rotation about the origin of the rectangle ABCD. 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
90^(∘)	(x,y)→ (- y,x)
180^(∘)	(x,y)→ (- x,- y)
270 ^(∘)	(x,y)→ (y,- x)

We want to rotate our rectangle ABCD 90^(∘) about the origin. Let's do it! ccc (x,y) & → & (- y, x) [0.5em] A(2,- 2) & → & A(2,2) [0.5em] B(4,- 2) & → & B(2,4) [0.5em] C(4,- 3) & → & C(3 ,4) [0.5em] D(2,- 3) & → & D(3,2) We can draw the rectangle after the rotation!

Next, we are asked to reflect rectangle ABCD across the x-axis.

Finally, we will translate rectangle ABCD 6 units to the left and 6 units up.

We found that Sequence C maps ABCD to A'B'C'D'.

Sequence D

In this sequence, we will first translate rectangle ABCD 6 units left.

Next, let's reflect the rectangle across the y-axis.

Now, we will translate rectangle ABCD 6 units down.

Finally, we will rotate the rectangle 180^(∘) about the origin.

We can see that Sequence D does not map rectangle ABCD to A'B'C'D'. Let's summarize our results.

Sequence	Does it map ABCD 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 ABCD to A'B'C'D'.

We will describe an example of a sequence of transformations that maps A'B'C'D' onto ABCD.

First, let's rotate A'B'C'D' by 270^(∘) about the origin. To perform a rotation, 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
90^(∘)	(x,y)→ (- y,x)
180^(∘)	(x,y)→ (- x,- y)
270 ^(∘)	(x,y)→ (y,- x)

We want to rotate our rectangle A'B'C'D' 270^(∘) about the origin. Let's do it! ccc (x,y) & → & ( y, - x) [0.5em] A'(-4,4) & → & A'(4,4) [0.5em] B'(- 4,2) & → & B'(2,4) [0.5em] C'(-3,2) & → & C'(2,3) [0.5em] D'(- 3,4) & → & D'(4,3) We can now plot the obtained points and draw the image of the rectangle after the rotation!

We can see that the rectangle A'B'C'D is right above ABCD, but the vertices do not match! To make them match, we can reflect A'B'C'D across the line that cuts the rectangle in the middle. This line can be expressed with an equation x=3.