The transformation (r_((90^(∘),O))∘ R_(y=1))(△XYZ) can be written as r_((90^(∘),O))(R_(y=1)(△XYZ)). This means that △XYZ will be reflected in the line y=1 and then the image will be rotated 90^(∘) about the origin O.
X''(2,3) Y''(-1,2) Z''(0,-2)
Practice makes perfect
The transformation (r_((90^(∘),O))∘ R_(y=1))(△XYZ) can be written as r_((90^(∘),O))(R_(y=1)(△XYZ)). This means that we will reflect △XYZ in the line y=1 and then rotate △XYZ 90^(∘) about the origin O. Let's begin by drawing the given figure.
We will do these transformations one at a time.
Reflection
Let's start by reflecting △XYZ using the line y=1 as the line of reflection. We will label this image △X'Y'Z'.
Rotation
Let's now rotate △X'Y'Z' 90^(∘) about the origin. Unless we are told otherwise, we perform rotations counterclockwise. We will label the new image △X''Y''Z''.
Final Image
Finally, we will show the preimage △XYZ and the image △X''Y''Z'' after the reflection and the rotation.
As we can see, X''(2,3), Y''(-1,2), and Z''(0,-2) are the coordinates of the image of △ XYZ.
