The transformation T_(<-3,4>)∘ r_((90^(∘),O))(△ ABC) can be written as T_(<-3,4>)(r_((90^(∘),O))(△ ABC)). This means that △ABC will be rotated 90^(∘) about the origin O and then the image will be translated 3 units to the left and 4 units up.
A''(-5,6), B''(-4,3), C''(-2,5)
Practice makes perfect
The transformation T_(<-3,4>)∘ r_((90^(∘),O))(△ ABC) can be written as T_(<-3,4>)(r_((90^(∘),O))(△ ABC)). This means that △ABC will be rotated 90^(∘) about the origin O and then the image will be translated 3 units to the left and 4 units up.
Now, we will do the transformations one at a time.
Rotation
Let's do the rotation. We will label the image △A'B'C'. For a rotation of 90^(∘) about the origin, we can find the image of each vertex by changing the sign of the y-coordinate and exchanging the x- and y-coordinates.
Preimage
Change Sign of y
Exchange x and y (Image)
A( 2, 2)
( 2, -2)
A'( -2, 2)
B( -1, 1)
( -1, -1)
B'( -1, -1)
C( 1, -1)
( 1, 1)
C'( 1, 1)
The rotation can be seen as follows.
Translation
Let's now translate △A'B'C'. We will label the image △A''B''C''. We can find the image of each vertex by subtracting3 to the x-coordinate and adding4 to the y-coordinate.
Preimage
Add and Subtract
Simplify (Image)
A'(-2,2)
(-2 - 3,2 + 4)
A''(-5,6)
B'(-1,-1)
(-1 - 3,-1 + 4)
B''(-4,3)
C'(1,1)
(1 - 3,1 + 4)
C''(-2,5)
The translation can be seen as follows.
Final Image
Finally, we will show the preimage △ABC and the image △A''B''C'' after the rotation followed by the translation.
The vertices of △A''B''C'' are A''(-5,6), B''(-4,3), and C''(-2,5).
