Transformations in the Coordinate Plane

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Transformations in the Coordinate Plane
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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• TRANSFORMATIONS
• PART 1
• INTRODUCTION AND VOCABULARY

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Objectives
Identify reflections, rotations, and
translations. Graph transformations in the
coordinate plane. Identify and draw dilations.
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Vocabulary
transformation reflection preimage
rotation image translation center of dilation
reduction enlargement isometry
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The Alhambra, a 13th-century palace in Grenada,
Spain, is famous for the geometric patterns that
cover its walls and floors. To create a variety
of designs, the builders based the patterns on
several different transformations.
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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. A transformation maps the
preimage to the image. Arrow notation (?) is used
to describe a transformation, and primes () are
used to label the image.
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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
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.
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An isometry is a transformation that does not
change the shape or size of a figure.
Reflections, translations, and rotations are all
isometries. Isometries are also called congruence
transformations or rigid motions.
Recall that a reflection is a transformation that
moves a figure (the preimage) by flipping it
across a line. The reflected figure is called the
image. A reflection is an isometry, so the image
is always congruent to the preimage.
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A dilation is a transformation that changes the
size of a figure but not the shape. The image and
the preimage of a figure under a dilation are
similar.
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A dilation enlarges or reduces all dimensions
proportionally. A dilation with a scale factor
greater than 1 is an enlargement, or expansion. A
dilation with a scale factor greater than 0 but
less than 1 is a reduction, or contraction. We
will discuss dilations in an extra lesson at the
end of Chapter 8, and again in Chapter 10 when we
learn about similarity.
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• TRANSFORMATIONS
• PART 2
• TRANSFORMATIONS IN THE COORDINATE PLANE

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Measure translations in the coordinate plane in
terms of the coordinates, not a linear
measurement such as inches or centimeters.
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If the angle of a rotation in the coordinate
plane is not a multiple of 90, you can use sine
and cosine ratios to find the coordinates of the
image.
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• TRANSFORMATIONS
• PART 3
• COMPOSITION OF TRANSFORMATIONS

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Objectives
Apply theorems about isometries. Identify and
draw compositions of transformations, such as
glide reflections.
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Vocabulary
composition of transformations glide reflection
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A composition of transformations is one
transformation followed by another. For example,
a glide reflection is the composition of a
translation and a reflection across a line
parallel to the translation vector.
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The image after each transformation is congruent
to the previous image. By the Transitive Property
of Congruence, the final image is congruent to
the preimage. This leads to the following theorem.
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Example 1B Drawing Compositions of Isometries
Draw the result of the composition of isometries.
?KLM has vertices K(4, 1), L(5, 2), and M(1,
4). Rotate ?KLM 180 about the origin and then
reflect it across the y-axis.
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Example 1B Continued
Step 1 The rotational image of (x, y) is (x,
y).
K(4, 1) ? K(4, 1), L(5, 2) ? L(5, 2), and
M(1, 4) ? M(1, 4).
Step 2 The reflection image of (x, y) is (x, y).

K(4, 1) ? K(4, 1), L(5, 2) ? L(5, 2), and
M(1, 4) ? M(1, 4).
Step 3 Graph the image and preimages.
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Example 1B Continued
Question Could the composite transformation be
replaced by a single transformation?
Answer The reflection image of (x, y) is (x,
y). (A reflection across the x-axis.)
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Check It Out! Example 1
?JKL has vertices J(1,2), K(4, 2), and L(3, 0).
Reflect ?JKL across the y-axis and then rotate it
180 about the origin.
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Check It Out! Example 1 Continued
Step 1 The reflection image of (x, y) is (x, y).

Step 2 The rotational image of (x, y) is (x,
y).
Step 3 Graph the image and preimages.
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Check It Out! Example 3
Copy the figure showing the translation that maps
LMNP ? LMNP. Draw the lines of reflection
that produce an equivalent transformation.
LMNP ? LMNP
translation