Geometry Chapter 12 – Extending Transformational Geometry
Lesson 1 – Reflections
Learning Targets
Success Criteria
•
•
LT12-1: Identify, draw, and solve problems
involving reflections.
•
•
Identify reflections.
Draw reflections using compass and
protractor.
Reflect figures in the coordinate plane.
Determine the orientation of a figure.
Transformation:
Isometry (congruence transformation):
Isometries studied in this chapter:
1.
3.
2.
4.
Pre-Image:
Image:
Photographers often use reflections to add tension and more interesting composition to their
photographs. Notice that the distance a point lies above the water line appears to be the same as the
distance its image lies below the water.
Ex#1: Identify which transformations are reflections.
A.
B.
C.
D.
Page 1
Ex#2: Draw a reflection.
1. Place your protractor so that its 90° mark and
the center of the protractor are on the line, m.
A. Draw rn(P) =
m
2. Slide the protractor along m so that the base
line of the protractor (the line through the 0° and
180° marks) goes through P.
P
3. Measure the distance from P to m along the
base line. You may wish to draw this line lightly.
4. Locate P' on the other side of m along the base
line, the same distance from m.
B. Draw rm(ΔABC) =
C. Draw r l (ABCD) =
C
C
B
A
D
B
A
Ex#3: Reflect figures in the coordinate plane.
A. rx-axis(ΔXYZ) with X(2, -1), Y(-4, -3), Z(3, 2).
B. ry=x(ΔRST) with R(-2, 2), S(5, 0), T(3, -1)
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Ex#3 (cont.) Reflect figures in the coordinate plane.
∆JKL has vertices J(0, 3), K(-2, -1), and L(-6,1). Graph ΔJKL and its image in the lines:
C. x = - 4
D. y = 2
Orientation: the order of the designation of the vertices of a polygon. Orientation can be either
clockwise or counterclockwise.
Ex#4: Give the orientation of each figure.
A. quadrilateral QUAD
B. quadrilateral QUAD
T
U
D
C. ΔTRI and ΔT'R'I'
T'
Q
Q
A
A
I
R
U
D
Notice in C. above ΔTRI and ΔT'R'I' have opposite orientation.
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I'
R '
Lesson 2 – Translations
Learning Targets
Success Criteria
•
•
LT12-2: Identify, draw, and solve problems
involving translations.
Identify translations.
Draw a translation.
Ex#1: Identify translations. Tell whether each transformation appears to be a translation.
A.
B.
C.
D.
Ex#2: Draw translations.
A. Draw the translation along v' .
B. Draw the translation along w
' .
Page 4
Recall that a vector in the coordinate plane can be written
as 〈a , b〉 , where a represents the horizontal change and
b is the vertical change from the vector's tip to its tail.
CD is represented by the ordered pair 〈3,−4 〉 .
D(3, -4)
Ex#3: Draw translations in the coordinate plane.
A. Translate the triangle with vertices D(-3, -1),
B. Translate the quadrilateral with vertices R(2, 5),
E(2, -3), and F(-2, -2) along the vector 〈 4,−3〉 . S(0, 2), T(1,–1), and U(3, 1) along the vector
〈−3,−3〉 .
C. Give the image of the vertices of ∆DEF after a D. After a translation, the image of A(-8, -1) is
translation along the vector 〈 3,−1〉 .
A'(9, -5). What is the image of the point (5, -4)
after this same translation?
D(-3, -1), E(5, -3), and F(-2, -2)
Page 5
Lesson 3 – Rotations
Learning Targets
LT12-3: Identify, draw, and solve
problems involving rotations.
Success Criteria
•
•
•
Identify rotations.
Draw rotations.
Draw rotations in the coordinate
plane.
Magnitude:
Direction: (of rotation) can be either counterclockwise or
clockwise. From this point forward, all positive rotations
will be counterclockwise, and all negative rotations will be
clockwise.
counterclockwise
clockwise
Ex#1: Identify rotations. Tell whether each transformation appears to be a rotation.
A.
B.
C.
D
Ex#2:
A. Draw a +140° rotation of ΔABC about point K.
C
B
K
Page 6
A
B. Draw a -45° rotation of ∆TRI about the point T.
BY 270º ABOUT THE ORIGIN:
(x, y)
Page 7
Ex#3: Draw rotations in the coordinate plane.
A. ∆PQR has vertices P(1, 1), Q(4, 5), and
R(5, 1). Graph ΔPQR and its image after a
rotation 90° about the origin.
B. ∆JKL is shown below. What is the image of
point J after a rotation 270° counterclockwise
about the origin?
Page 8
Lesson 4 – Composition of Transformations
Learning Targets
Success Criteria
•
•
LT12-4: Identify, draw, and solve problems
involving the composition of isometric
transformations.
•
Draw compositions of isometries.
Describe transformations in terms of
reflections.
Draw a glide reflection.
composition of transformations:
Ex#1: Draw a composition of isometries.
A. ∆KLM has vertices K(4, –1), L(5, –2), and M(1, –4).
B. ∆JKL has vertices J(1, -1), K(4, -1), and L(3, 2). Reflect
Rotate ∆KLM 180° about the origin and then reflect it across ∆JKL across the x-axis and then translate it along the vector
the y-axis.
〈−3, 2 〉 .
Page 9
Ex#2: Describe transformations in terms of reflections.
Draw each image and describe a SINGLE transformation that maps the preimage onto the image.
A. rn(rm(ABCD))
B. rx = 3 ○ rx = -2(ΔEFG).
G
n
m
F
A
B
E
D
C
Ex#3: Describe transformations in terms of reflections.
Draw each image and describe a SINGLE transformation that maps the preimage onto the image.
A. rl(rm(ABCD))
B. rx = 3 ○ ry = -1(ΔFUN).
F
U
N
2 2(
C
B
l
A
m
A glide reflection is the composition of a translation followed by a reflection in a line parallel to the
translation vector.
Page 10
Ex#4: Draw a glide reflection.
∆ JKL has vertices J(6, -1), K(10, -2), and L(5, -3). Graph ΔJKL and its image after a translation along
〈0, 4〉 followed by reflection in the y-axis.
Lesson 5- Symmetry
Learning Targets
Success Criteria
•
•
•
LT12-5: Identify, describe, and solve problems
involving geometric symmetry.
Identify line symmetry.
Identify rotational symmetry.
Identify symmetry in 3-D.
In the animal kingdom, the symmetry of an animal's
body is often an indication of the animal's complexity.
Animals displaying line symmetry, such as insects, are
usually more complex life forms than those displaying
rotational symmetry, like a jellyfish.
Ex#1: Identify line symmetry. Draw all lines that seem to be symmetry lines.
I.
II.
III.
IV.
V.
VI.
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VII.
VIII.
IX.
Rotational Symmetry
A figure in the plane has rotational symmetry (or
radial symmetry) if the figure can be mapped onto
itself by a rotation between 0° and 360° about the
center of the figure, called the center of
symmetry (or point of symmetry).
order of symmetry: the number of times a figure
maps onto itself as it rotates from 0° and 360°
magnitude of symmetry: (angle of rotation) is
the smallest angle through which a figure can be
rotated so that it maps onto itself.
magnitude=
360 °
order
Page 12
Ex#2: Identify rotational symmetry.
A. Determine whether each figure has rotational symmetry.
B. If the figure has rotational symmetry, label the center C, and state the order and magnitude of
symmetry.
I.
II.
III.
IV.
V.
VI.
VII.
IX.
VIII.
Ex#3: Identify 3-D symmetry. Tell whether each figure has plane symmetry, symmetry about an
axis, or neither.
A.
B.
C.
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D.
Lesson 7 – Dilations
Learning Targets
Success Criteria
•
•
•
LT12-7: Identify, draw, and solve problems
involving dilations.
•
Identify dilations.
Draw a dilation.
Find the scale factor and center for a
dilation.
Draw a dilation in the coordinate plane.
Dilation:
Ex#1: Identify dilations. Tell whether each transformation appears to be a dilation.
A.
B.
C.
Dilation
A dilation with center C and positive scale factor
k, k ≠ 1, maps a point P in a figure to its image
such that
● if point P and C coincide, then the image
and preimage are the same point, or
● if point P is not the center of dilation, then
P' lies on '
CP and CP' = k(CP).
Page 14
Ex#2: Draw a dilation.
Use a ruler to draw the image of ΔABC under a dilation with center D and scale factor
1
.
2
B
D
C
A
Ex#3: Draw a dilation.
Use a ruler to draw the image of JKLM under a dilation with center N and scale factor
K
J
L
N
M
Notice:
● if k > 1, then the dilation is a(n)
.
● if 0 < k < 1, then the dilation is a(n)
.
● if k = 1, then the dilation is called the
.
● if k = -1, then the dilation is a
.
Page 15
3
.
2
Ex#3: Find the scale factor and center of each dilation. Determine if it is an enlargement or
reduction.
U '
I
B.
A.
K
A '
Q '
T
U
A
I '
Q
D '
K '
T '
D
E
E '
If the scale factor of a dilation is negative, the
preimage is rotated by 180°. For k > 0, a
dilation with a scale factor of –k is equivalent
to the composition of a dilation with a scale
factor of k that is rotated 180° about the
center of dilation.
Ex#4: Draw dilations in the coordinate plane.
Draw the image of the triangle with vertices
P(–4, 4), Q(–2, –2), and R(4, 0) under a dilation
with a scale factor of ½ centered at the origin.
Page 16
C