Recall that a counterclockwise rotation of n ^(∘) is the same as a clockwise rotation of (360-n) ^(∘). Similarly, a clockwise rotation of n^(∘) is the same as a counterclockwise rotation of (360-n) ^(∘).
We want to determine which questions ask for different types of transformation. To do so, let's remember a remark about the relationship between a clockwise and counterclockwise rotations.
A counterclockwise rotation of n ^(∘) is the same as a clockwise rotation of (360-n) ^(∘). Similarly, a clockwise rotation of n^(∘) is the same as a counterclockwise rotation of (360-n) ^(∘).
Let's take a look at the first question.
It describes a 90 ^(∘) clockwise rotation. From the remark, we know that a clockwise rotation of n^(∘) is the same as a counterclockwise rotation of (360-n) ^(∘). In this case n = 90. Let's find (360-n)^(∘).
(360- 90)^(∘) = 270 ^(∘)
This result tells us that the first question is the same as asking for the coordinates of the image after a 270^(∘) counterclockwise rotation about the origin. We can tell that is is exactly for what the fourth question asks.
Now, let's take a look at the second question.
Following the same reasoning as before, we know that a clockwise rotation of n^(∘) is the same as a counterclockwise rotation of (360-n) ^(∘). In this case n = 270. Let's find (360-n)^(∘).
(360- 270)^(∘) = 90 ^(∘)
This means that the second question is the same as asking for the coordinates of the image after a 90 ^(∘) counterclockwise rotation about the origin. Let's take a look if this is for what the third question asks.
Note that turning the figure to the right is the same as turning the figure clockwise about the origin. This means it is not the same as for what the second question asks. However, it is the same as the first question and the fourth one. The first, third, and fourth question ask for the same thing, and the second question asks for a different thing.
Finding the Coordinates
Now, let's find the coordinates of the image in those two situations. Recall that when a point (x,y) is rotated counterclockwise about the origin, the following are true.
For a rotation of 90 ^(∘), (x,y) → (- y, x).
For a rotation of 180 ^(∘), (x,y) → (- x, - y).
For a rotation of 270 ^(∘), (x,y) → (y, - x).
As we already know, the first, third, and fourth question ask for the same thing. We will use the information given in the fourth question, because it describes the counterclockwise transformation.
We know that for a rotation of 270 ^(∘), the coordinates (x,y) change to (y, - x). Let's identify the coordinates of the given image and find its coordinates after this rotation.
(x,y)
(y, - x)
A(2,4)
A'(4, -2)
B(4,4)
B'(4, -4))
C(4,1)
C'(1,-4)
Now that we found the coordinates of the image after a 270 ^(∘) counterclockwise rotation about the origin, let's graph it on a coordinate plane.
Now we can find the coordinates of the image that the second question asks for.
We already found that it is the same as asking for the coordinates of the image after a 90 ^(∘) counterclockwise rotation about the origin. We know that it means that the coordinates (x,y) will become (- y, x). Let's find them!
(x,y)
(- y, x)
A(2,4)
A''(- 4, 2)
B(4,4)
B''(- 4, 4))
C(4,1)
C''(-1,4)
Finally, we can plot the figure after this rotation.
