Chapter 9: Properties of Transformations Guided...
Transcript of Chapter 9: Properties of Transformations Guided...
Geometry Winter Semester
Name: ______________
Chapter 9: Properties of Transformations
Guided Notes
CH. 9 Guided Notes, page 2
9.1 Translate Figures and Use Vectors
Term Definition Example
transformation
image
preimage
translation
isometry
Theorem 9.1 Translation Theorem
A translation is an isometry.
vector
Vectors
1. Initial Point- 2. Terminal Point- 3. Horizontal Component-
4. Vertical Component-
component form
CH. 9 Guided Notes, page 3 Examples: 1. Graph quadrilateral ABCD with vertices A(-2,6), B(2,4), C(2,1), and D(-2,3). Find the image of each vertex after the translation (x,y)
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" (x+3 , y-3). The graph the image using prime notation. 2. Write a rule for the translation of
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"ABC to
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"A’B’C’. Then verify that the translation is an isometry.
CH. 9 Guided Notes, page 4 3. Name the vector and write it’s component form.
a) b)
4. The vertices of
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"ABC are A(0,4), B(2,3) and C(1,0). Translate
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"ABC using the vector
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"4,1 .
CH. 9 Guided Notes, page 5
9.2 Use Properties of Matrices
Term Definition Example
matrix (matrices)
element
dimensions
adding and subtracting matrices
The matrices must have the same dimensions.
image matrix
multiplying matrices
CH. 9 Guided Notes, page 6
9.3 Perform Reflections
Term Definition Example
reflection
line of reflection
Theorem 9.2 Reflection Theorem
A reflection is an isometry. Case 1: Case 2: Case 3: Case 4:
Coordinate Rules for Reflections 1. If (a,b) is reflected in the x-axis, its image is the point (a,-b). 2. If (a,b) is reflected in the y-axis, its image is the point (-a,b). 3. If (a,b) is reflected in the line y = x, its image is the point (b,a). 4. If (a,b) is reflected in the line y = -x, its image is the point (-b,-a).
CH. 9 Guided Notes, page 7 Examples:
1. The vertices of
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"ABC are (1,2), B(3,0), and C(5,3). Graph the reflection of
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"ABC as described.
a) In the line x = 2. b) In the line y = 3 2. The endpoints of
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CD are C(-2,2) and D(1,2). Reflect the segment in the line y=x. Graph the segment and its image. 3. Reflect
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CD from example 2 in the line y = -x. Graph
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CD and its image.
CH. 9 Guided Notes, page 8
9.4 Perform Rotations
Term Definition Example
rotation
center of rotation
angle of rotation
direction of
rotation
rotations about
the origin
Theorem 9.3 Rotation Theorem
A rotation is an isometry. Case 1: Case 2: Case 3:
CH. 9 Guided Notes, page 9 Coordinate Rules for Rotations about the Origin
When a point (a,b) is rotated counterclockwise about the origin, the following are true:
1. For a rotation of 90°, (a,b) (-b,a).
2. For a rotation of 180°, (a,b) (-a,-b).
3. For a rotation of 270°, (a,b) (b,-a).
Examples: 1. Draw a
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150° rotation of
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"ABC about
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P .
2. Graph quadrilateral KLMN with vertices K(3,2), L(4,2), M(4,-3), N(2,-1). Then rotate the quadrilateral
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270° counterclockwise about the origin.
CH. 9 Guided Notes, page 10 3. The quadrilateral is rotated about P. Find the value of y.
CH. 9 Guided Notes, page 11
9.5 Apply Compositions of Transformations
Term Definition Example
glide reflection
composition of transformations
Theorem 9.4 Composition Theorem
The composition of two (or more) isometries is an isometry.
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.
Theorem 9.6 Reflections in Intersecting
Lines Theorem
If lines k and m intersect at point P, then a reflection in line k followed by a reflection in line m is the same as a rotation about point P.
CH. 9 Guided Notes, page 12 Examples: 1. The vertices of
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"ABC are A(2,1), B(5,3), and C(6,2). Find the image of
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"ABC after the glide reflection. Translation: (x,y)
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" (x-8, y) Reflection: in the x-axis
2. The endpoints of
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CD are C(-2,6) and D(-1,3). Graph the image of
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CD after the composition.
Reflection: in the y-axis Rotation:
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90° about the origin
3. In the diagram, a reflection in line
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k maps
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GF to G’F’. A reflection in line
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m maps G’F’ to G’’F’’. Also
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FA = 6 and
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DF’ = 3. a) Name any segments congruent to
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GF ,
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FA, and
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GB. b) What is the length of GG’’?
CH. 9 Guided Notes, page 13 4 In the diagram, the figure is reflected in line
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k . The image is then reflected in line
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m . Describe a single transformation that maps F to F’’.
CH. 9 Guided Notes, page 14
9.6 Identify Symmetry
Term Definition Example
line symmetry
line of
symmetry
rotation symmetry
center of symmetry
angle of rotation
Examples: 1. How many lines of symmetry does the figure have? a) b) c)
CH. 9 Guided Notes, page 15 2. Does the figure have rotational symmetry? If so, describe any rotations that map
the figure onto itself (including the degree of rotation). a) b) c)
3. Identify any lines of symmetry and any rotational symmetry of the figure. More Practice 4. Describe any lines of symmetry and rotational symmetry of the figures.
CH. 9 Guided Notes, page 16
9.7 Identify and Perform Dilations
Term Definition Example
dilation
similarity
transformation
center of dilation
scale factor of a
dilation
reduction
enlargement
Coordinate
Notation for a Dilation
matrices—
scalar multiplication