The given composition of isometries represents a glide reflection.
R_(x=0)∘ T_(<0,5>)
Reflection across x=0
Translation 5 units up
To complete a glide reflection, we first perform the translation and then the reflection.
Translation
Let's first start with identifying the coordinates of the given figure.
To move ABCD five units up, we have to add 5 to each y-coordinate.
(x,y) → (x,y+5)
Let's use a table of values to show the changes in the y-coordinate.
(x,y)
(x,y+ 5)
(x',y')
A(3,0)
(3, 0+ 5)
A'(3,5)
B(1,- 2)
(1,- 2+ 5)
B'(1,3)
C(3,- 5)
(3,- 5+ 5)
C'(3,0)
D(7,- 1)
(7,- 1+ 5)
D'( 7,4)
With these points, we are able to draw the translated polygon.
Reflection
To complete the reflection, we have to move all of the vertices of ABCD to the opposite side of the y-axis in a way such that the distance from the vertices to the y-axis remains the same. Note that a reflection in the y-axis switches the sign of the x-coordinate, and does not affect the y-coordinate. Therefore, the image of the reflection of a point (a,b) in the y-axis is (- a,b).
Final Glide Reflection
The final glide reflection is the combined translation and reflection.
As we can see, A''(-3,5), B''(-1,3), C''(-3,0), and D''(-7,4) are the coordinates of the image of ABCD.
