The transformation R_(y=1)∘ T_(<2,-1>)(△ ABC) can be written as R_(y=1)(T_(<2,-1>)(△ ABC)). This means that △ABC will be translated 2 units to the right and 1 unit down. Then, the image will be reflected using y=1 as the line of reflection.
See solution.
Practice makes perfect
The transformation R_(y=1)∘ T_(<2,-1>)(△ ABC) can be written as R_(y=1)(T_(<2,-1>)(△ ABC)). This means that △ABC will be translated 2 units to the right and 1 unit down. Then, the image will be reflected using y=1 as the line of reflection.
Now, we will do the transformations one at a time.
Translation
Let's do the translation. We will label the image △A'B'C'. We can find the image of each vertex by adding2 to the x-coordinate and subtracting1 from the y-coordinate.
Preimage
Add and Subtract
Simplify (Image)
A(2,2)
(2 + 2,2 - 1)
A'(4,1)
B(-1,1)
(-1 + 2,1 - 1)
B'(1,0)
C(1,-1)
(1 + 2,-1 - 1)
C'(3,-2)
The translation can be seen as follows.
Reflection
Let's now reflect △A'B'C' using y=1 as the line of reflection, one vertex at a time. We will label the image △A''B''C''. First, we will reflect the point A'(4,1). To do this, let's begin by drawing the line of reflection.
Since the point A'(4,1) is on the line of reflection, the image after the reflection will be in the same place. Let's now reflect the point B'(1,0). To do so, we will draw a line that is perpendicular to the reflection line and passes through B'(1,0). A detailed explanation on how to draw this line can be seen here.
Then, we can measure the distance from the preimage to the line of reflection and locate the image the same distance from the given line, but on the opposite side.
We can repeat the procedure to find C''.
Let's finally connect the obtained points to draw △ A''B''C''.
The vertices of △ A''B''C'' are A''(4,1), B''(1,2), and C''(3,4). Let's have a look at the reflection.
Final Image
Finally, we will show the preimage △ABC and the image △A''B''C'' after the rotation and the translation.
