Identify reflections, rotations, and translations.

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Objectives
Identify reflections, rotations, and
translations. Graph transformations in the
coordinate plane.
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Vocabulary
transformation reflection preimage
rotation image translation
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A transformation is a change in the position,
size, or shape of a figure. The original figure
is called the preimage. The resulting figure is
called the image.
Arrow notation (?) is used to describe a
transformation, and primes () are used to label
the image.
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Example 1A Identifying Transformation
Identify the transformation. Then use arrow
notation to describe the transformation.
The transformation cannot be a reflection because
each point and its image are not the same
distance from a line of reflection.
90 rotation, ?ABC ? ?ABC
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Example 1B Identifying Transformation
Identify the transformation. Then use arrow
notation to describe the transformation.
The transformation cannot be a translation
because each point and its image are not in the
same relative position.
reflection, DEFG ? DEFG
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Check It Out! Example 1
Identify each transformation. Then use arrow
notation to describe the transformation.
a.
b.
translation MNOP ? MNOP
Counterclockwise rotation ?XYZ ? ?XYZ
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Example 2 Drawing and Identifying Transformations
A figure has vertices at A(1, 1), B(2, 3), and
C(4, 2). After a transformation, the image of
the figure has vertices at A'(1, 1), B'(2, 3),
and C'(4, 2). Draw the preimage and image. Then
identify the transformation.
Plot the points. Then use a straightedge to
connect the vertices.
The transformation is a reflection across the
y-axis because each point and its image are the
same distance from the y-axis.
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Check It Out! Example 2
A figure has vertices at E(2, 0), F(2, -1), G(5,
-1), and H(5, 0). After a transformation, the
image of the figure has vertices at E(0, 2),
F(1, 2), G(1, 5), and H(0, 5). Draw the
preimage and image. Then identify the
transformation.
Plot the points. Then use a straightedge to
connect the vertices.
The transformation is a 90 counterclockwise
rotation with rotation center at origin O(0,0).
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What happens when we translate a shape ? The
shape remains the same size and shape and the
same way up it just. .
Transformations
slides
3. Translation
Horizontal translation
Write the rule to describe a translation from..










Vertical translation
D
C
A
B
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Writing a Rule for a translation
y
To write a rule, look for the change in the x and
y values for a coordinate.










B
From A to A. The point has gone 3 units to the
right and 2 units up.
B
A
C
The rule is (x,y) ? (x ?), (y ?)
C
A
The rule is (x,y) ? (x 3), (y 2)
x
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Your turn, Write the Rule.
y










The rule is (x,y) ? (x ?), (y ?)
The rule is (x,y) ? (x -4), (y -4)
x
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Your turn, Write the Rule.
y










The rule is (x,y) ? (x ?), (y ?)
The rule is (x,y) ? (x 1), (y - 5)
x
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Your turn, Write the Rule.
y










The rule is (x,y) ? (x ?), (y ?)
The rule is (x,y) ? (x - 5), (y - 2)
x
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Example 3 Translations in the Coordinate Plane
Find the coordinates for the image of ?ABC after
the translation (x, y) ? (x 2, y - 1). Draw
the image.
Step 1 Find the coordinates of ?ABC. The
vertices of ?ABC are A(4, 2), B(3, 4), C(1,
1).
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Example 3 Continued
Step 2 Apply the rule (x, y) ? (x 2, y - 1) to
find the vertices of the image. A(4 2, 2 1)
A(2, 1) B(3 2, 4 1) B(1, 3) C(1
2, 1 1) C(1, 0)
Step 3 Plot the points. Then finish drawing the
image by using a straightedge to connect the
vertices.
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To find coordinates for the image of a figure in
a translation, add a to the x-coordinates of the
preimage and add b to the y-coordinates of the
preimage. Translations can also be described by a
rule such as (x, y) ? (x a, y b).
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Check It Out! Example 3
Find the coordinates for the image of JKLM after
the translation (x, y) ? (x 2, y 4). Draw the
image.
Step 1 Find the coordinates of JKLM. The
vertices of JKLM are J(1, 1), K(3, 1), L(3, 4),
M(1, 4), .
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Check It Out! Example 3 Continued
Step 2 Apply the rule to find the vertices of the
image. J(1 2, 1 4) J(1, 5) K(3 2, 1
4) K(1, 5) L(3 2, 4 4) L(1, 0) M(1
2, 4 4) M(1, 0)
Step 3 Plot the points. Then finish drawing the
image by using a straightedge to connect the
vertices.
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Example 4 Art History Application
The figure shows part of a tile floor. Write a
rule for the translation of hexagon 1 to hexagon
2.
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Example 4 Continued
The figure shows part of a tile floor. Write a
rule for the translation of hexagon 1 to hexagon
2.
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Check It Out! Example 4
Use the diagram to write a rule for the
translation of square 1 to square 3.
Step 1 Choose two points. Choose a Point A on
the preimage and a corresponding Point A on the
image. A has coordinate (3, 1) and A has
coordinates (1, 3).
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Check It Out! Example 4 Continued
Use the diagram to write a rule for the
translation of square 1 to square 3.
Step 2 Translate. To translate A to A, 4 units
are subtracted from the x-coordinate and 4
units are subtracted from the
y-coordinate. Therefore, the translation rule is
(x, y) ? (x 4, y 4).
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Lesson Quiz Part I
1. A figure has vertices at X(1, 1), Y(1, 4),
and Z(2, 2). After a transformation, the image
of the figure has vertices at X'(3, 2), Y'(1,
5), and Z'(0, 3). Draw the preimage and the
image. Identify the transformation.
translation
2. What transformation is suggested by the wings
of an airplane?
reflection
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Lesson Quiz Part II
4. Find the coordinates of the image of F(2, 7)
after the translation (x, y) ? (x 5, y 6).
(7, 1)
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Architecture Application
5. Is there another transformation that can be
used to create this frieze pattern? Explain your
answer.
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Objectives
Use properties of rigid motions to determine
whether figures are congruent.
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Vocabulary
Isometry Rigid transformation Dilation
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An isometry is a transformation that preserves
length, angle measure, and area. Because of these
properties, an isometry produces an image that is
congruent to the preimage.
A rigid transformation is another name for an
isometry. Reflection, rotation and translation
are isometry, or rigid transformation.
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A dilation with scale factor k gt 0 and center (0,
0) maps (x, y) to (kx, ky).
Dilation is not isometry. It is not a rigid
transformation.
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Example 5 Drawing and Identifying Transformations
M (x, y) ? (3x, 3y) K(-2, -1), L(1,
-1), N(1, -2))
dilation with scale factor 3 and center (0, 0)
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Check It Out! Example 5

1. Apply the transformation M (x, y) ?(3x, 3y) to
   the polygon with vertices D(1, 3), E(1, -2), and
   F(3, 0). Name the coordinates of the image
   points. Identify and describe the transformation.

D(3, 9), E(3, -6), F(9, 0) dilation with
scale factor 3
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Lesson Quiz Part-I
Apply the transformation M to the polygon with
the given vertices. Identify and describe the
transformation.
1. M (x, y) ? (3x, 3y) A(0, 1), B(2, 1), C(2, -1)
dilation with scale factor 3 and center (0, 0)
2. M (x, y) ? (-y, x) A(0, 3), B(1, 2), C(4, 5)
90 rotation counterclockwise with center of
rotation (0, 0)
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Lesson Quiz Part-II
3. M (x, y) ? (x 1, y - 2) A(-2, 1), B(-2, 4),
C(0, 3)
translation 1 unit right and 2 units down
4. Determine whether the triangles are congruent.
A(1, 1), B(1, -2), C(3, 0) J(2, 2), K(2, -4),
L(6, 0)
not ? ? ABC can be mapped to ? JKL by a dilation
with scale factor k ? 1 (x, y) ? (2x, 2y).
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Lesson Quiz Part-III
5. Prove that the triangles are congruent. A(1,
-2), B(4, -2), C(1, -4) D(-2, 2), E(-5, 2), F(-2,
0)
? ABC can be mapped to ? A'B'C' by a translation
(x, y) ? (x 1, y 4) and then ? A'B'C' can be
mapped to ?DEF by a reflection (x, y) ? (-x, y).