Holt Geometry

1-7 Transformations in the Coordinate Plane1-7 Transformations in the Coordinate Plane

Holt Geometry

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Holt Geometry

1-7 Transformations in the Coordinate Plane

TRANSFORMATIONS

PART 1

INTRODUCTION AND VOCABULARY

Holt Geometry

1-7 Transformations in the Coordinate Plane

Identify reflections, rotations, and translations.

Graph transformations in the coordinate plane.

Identify and draw dilations.

Objectives

Holt Geometry

1-7 Transformations in the Coordinate Plane

transformation reflectionpreimage rotationimage translationcenter of dilation reductionenlargement isometry

Vocabulary

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

Holt Geometry

1-7 Transformations in the Coordinate Plane

Holt Geometry

1-7 Transformations in the Coordinate Plane

Example 1B: Identifying Transformation

Identify the transformation. Then use arrow notation to describe the transformation.

The transformation cannot bea translation because eachpoint and its image are not inthe same relative position.

reflection, DEFG D’E’F’G’

Holt Geometry

1-7 Transformations in the Coordinate Plane

Check It Out! Example 1

Identify each transformation. Then use arrow notation to describe the transformation.

translation; MNOP M’N’O’P’ rotation; ∆XYZ ∆X’Y’Z’

a. b.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

Holt Geometry

1-7 Transformations in the Coordinate Plane

For a dilation with scale factor k, if k > 0, the figure is not turned or flipped. If k < 0, the figure is rotated by 180°.

Helpful Hint

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

TRANSFORMATIONS

PART 2

TRANSFORMATIONS IN THE COORDINATE PLANE

Holt Geometry

1-7 Transformations in the Coordinate Plane

Holt Geometry

1-7 Transformations in the Coordinate Plane

Measure translations in the coordinate plane in terms of the coordinates, not a linear measurement such as inches or centimeters.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

TRANSFORMATIONS

PART 3

COMPOSITION OF TRANSFORMATIONS

Holt Geometry

1-7 Transformations in the Coordinate Plane

Apply theorems about isometries.

Identify and draw compositions of transformations, such as glide reflections.

Objectives

Holt Geometry

1-7 Transformations in the Coordinate Plane

composition of transformationsglide reflection

Vocabulary

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

The glide reflection that maps ∆JKL to ∆J’K’L’ is the composition of a translation along followed by a reflection across line l.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

K

L

M

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

K

L

M

M’

K’

L’ L”

M”

K”

Holt Geometry

1-7 Transformations in the Coordinate Plane

Example 1B Continued

Question: Could the composite transformation be replaced by a single transformation?

K

L

M

M’

K’

L’ L”

M”

K”

Answer: The reflection image of (x, y) is (x, –y). (A reflection across the x-axis.)

Holt Geometry

1-7 Transformations in the Coordinate Plane

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.

L

KJ

Holt Geometry

1-7 Transformations in the Coordinate Plane

LKJ

L'’

J’K’

K”J”

L'

Check It Out! Example 1 Continued

Step 2 The rotational image of (x, y) is (–x, –y).

J’(–1, –2) J”(1, 2), K’(–4, –2) K”(4, 2), and L’(–3, 0) L”(3, 0).

Step 1 The reflection image of (x, y) is (–x, y).

J(1, –2) J’(–1, –2), K(4, –2) K’(–4, –2), and L(3, 0) L’(–3, 0).

Step 3 Graph the image and preimages.

Holt Geometry

1-7 Transformations in the Coordinate Plane

Holt Geometry

1-7 Transformations in the Coordinate Plane

Holt Geometry

1-7 Transformations in the Coordinate Plane

Check It Out! Example 3

Copy the figure showing the translation that maps LMNP L’M’N’P’. Draw the lines of reflection that produce an equivalent transformation.

LMNP L’M’N’P’translation:

L M

P N

L’ M’

P’ N’

Step 1 Draw MM’ and locate the midpoint X of MM’ X

Step 2 Draw the perpendicular bisectors of MX and M’X.