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1-7

Transformations in the Coordinate Plane

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

- TRANSFORMATIONS
- PART 1
- INTRODUCTION AND VOCABULARY

Objectives

Identify reflections, rotations, and

translations. Graph transformations in the

coordinate plane. Identify and draw dilations.

Vocabulary

transformation reflection preimage

rotation image translation center of dilation

reduction enlargement isometry

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.

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

Check It Out! Example 1

Identify each transformation. Then use arrow

notation to describe the transformation.

a.

b.

translation MNOP ? MNOP

rotation ?XYZ ? ?XYZ

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.

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.

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.

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.

- 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.

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.

- TRANSFORMATIONS
- PART 3
- COMPOSITION OF TRANSFORMATIONS

Objectives

Apply theorems about isometries. Identify and

draw compositions of transformations, such as

glide reflections.

Vocabulary

composition of transformations glide reflection

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.

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.

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.

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.)

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.

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