Transcript Chapter 9.5 Notes: Apply Compositions of Transformations
Chapter 9.5 Notes: Apply Compositions of Transformations Goal: You will perform combinations of two or more transformations.
• A
glide reflection
is a transformation in which every point P is mapped to a point P’’ by the following steps.
Step 1
: First, a translation maps P to P’.
Step 2
: Then, a reflection in a line
k
parallel to the direction of the translation maps P’ to P’’.
Ex.1: The vertices of ΔABC are A(3, 2), B(6, 3), and C(7, 1). Find the image of ΔABC after the glide reflection. Translation:
x y
(
x
12,
y
2)
Reflection: in the x-axis Ex.2: The vertices of ΔPQR are P(2, 6), Q(4, -2), and R(-3, -3). Find the image of ΔPQR after the glide reflection. Translation:
x y
(
x
4,
y
1)
Reflection: in the x-axis
• When two or more transformations are combined to form a single transformation, the result is a
composition of transformations
. •
Theorem 9.4 Composition Theorem:
The composition of two or more isometries is an isometry. Ex.3: ΔRST has vertices R(1, -3), S(2, -6), and T(5, -4). Graph the image of ΔRST after the composition. Reflection: in the y-axis Rotation: 90 o clockwise about the origin
Ex.4: ΔMNO has vertices M(-2, 1), N(3, 4), and O(1, 1). Graph the image of ΔMNO after the composition. Reflection: in the line y = -1 Rotation: 180 o clockwise about the origin
•
Theorem 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.
''.
Also, HB = 9 and DH’’ = 4.
a. Name any segment congruent to each segment: HG, HB, and GA b. Does AC = BD? Explain.
''?
'
Ex.6: Quadrilateral ABCD has vertices A(1, 4), B(5, 5), C(7, 2), and D(2, 2). Graph the image of Quadrilateral ABCD after the composition. Reflection: in the y-axis Rotation: 180 o clockwise about the origin Ex.7: ΔABC has vertices A(-4, 8), B(-6, 2), and C(-2, 2). Graph the image of ΔABC after the composition. Translation:
x y
(
x
4,
y
3)
Reflection: in the x – axis