1. 9.5 Apply Compositions of Transformations
Please have homework ready
Find the image of (2,3) under each transformation.
1. translation (x,y) => (x-6, y-2)
(-4,1)
2. Reflection in the y axis
(-2,3)
3. Reflection in the line y = 6
(2,9)
4. Rotation 90 degrees about the origin
(-3,2)

2. Vocab and Theorems
Glide reflections – a transformation in which every point P is mapped to a point P’’ by the following steps.
1. A translation maps P to P’.
2. A reflection in a line k parallel to the direction of the translation maps P’ to P’’
A composition of transformations – the result of two or more transformations that are combined to form a single transformation

3. The vertices of ABC are A(4,3), B(7,4) and C(8,2)
The vertices of ABC are A(4,3), B(7,4) and C(8,2). Find the image of ABC after the glide reflection.
Translation: (x,y) =>(x-11,y)
Reflection: in the x-axis

4. Theorems
Theorem 9.4 Composition Theorem – The composition of two (or more) isometries is an isometry.

5. Composition
The endpoints of RS and R(2,-4) and S(3,-7). Graph the image of RS after the composition.
Reflection in the y-axis
Rotation: 90 degrees about the origin

6. Theorems 9.5 Reflections in Parallel Lines Theorem
If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.
If P’’ is the image of P, then
1. PP’’ is perpendicular to k and m, and
2. PP’’ = 2d, where d is the distance between k and m

7. Apply
In the diagram, a reflection in line k maps GH to G’H’. A reflection in line m maps G’H’ to G’H’’. Also, HB = 10 and DH” = 5
a. Name any segments congruent to each segment HG, HB, GA.

8. Apply
In the diagram, a reflection in line k maps GH to G’H’. A reflection in line m maps G’H’ to G’H’’. Also, HB = 10 and DH” = 5
a. Name any segments congruent to each segment HG, HB, GA.
HG = H’G’,
HG = H’’G’’
HB = H’B
GA = G’A

9. Apply
In the diagram, a reflection in line k maps GH to G’H’. A reflection in line m maps G’H’ to G’H’’. Also, HB = 10 and DH” = 5
Does AC = BD?

10. Apply
In the diagram, a reflection in line k maps GH to G’H’. A reflection in line m maps G’H’ to G’H’’. Also, HB = 10 and DH” = 5
Does AC = BD?
Yes, AC = BD because G’G’’ and HH’ are perpendicular to both k and m, so BD and AC are opposite sides of a
rectangle.

11. Apply
In the diagram, a reflection in line k maps GH to G’H’. A reflection in line m maps G’H’ to G’H’’. Also, HB = 10 and DH” = 5
a. What is the length of GG’’?

12. Apply
In the diagram, a reflection in line k maps GH to G’H’. A reflection in line m maps G’H’ to G’H’’. Also, HB = 10 and DH” = 5
a. What is the length of GG’’?
By the properties of reflections, H’B = 10 and H’D = 5. Theorem 9.5 implies that
GG’’ = HH’’ = 2 *BD so
the length of GG’’ is
2(10+5), or 30 units

13. Quiz Prep (Friday) Graph reflections of the line, x and y axis
Rotation about the origin in 90, 180 and 270 degrees.
Compositions