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Chapter 9 Transformations

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4.8 Transformations An operation that moves or changes a geometric figure (a preimage) in some way to produce a new figure (an image).

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**Congruence transformations**

changes the position of the figure without changing the size or shape. Translation Reflection Rotation

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A Translation moves every point of a figure the same distance in the same direction. Coordinate notation: (x , y) (x + a, y + b)

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Example The vertices of ABC are A(4, 4), B(6, 6), and C(7, 4). The notation (x, y) → (x + 1, y – 3) describes the translation of ABC to DEF. What are the vertices of DEF?

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A Reflection Uses a line of reflection to create a mirror image of the original figure. Coordinate notation for reflection in the x-axis : (x ,y) (x , -y) Coordinate notation for reflection in the y- axis: (x , y) (-x, y)

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Example Reflect a figure in the x-axis

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Rotation Turns a figure about a fixed point called the center of rotation

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Examples Graph AB and CD. Tell whether CD is a rotation of AB about the origin. If so, give the angle and direction of rotation. A(–3, 1), B(–1, 3), C(1, 3), D(3, 1)

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**Tell whether PQR is a rotation of STR**

Tell whether PQR is a rotation of STR. If so, give the angle and direction of rotation.

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**Name the type of transformation demonstrated in each picture.**

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**Name the type of transformation shown.**

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6.7 Dilations A transformation that stretches or shrinks a figure to create a similar figure. A figure is reduced or enlarged with respect to a fixed point called the center of dilation.

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The scale factor of a dilation is the ratio of the side length of the image to the corresponding side length of the original figure Coordinate notation for a dilation with respect to the origin: (x ,y) ( kx, ky) Reduction: 0 < k < 1 Enlargement : k > 1

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Examples Draw a dilation of quadrilateral ABCD with vertices A(2, 1), B(4, 1), C(4, – 1), and D(1, – 1). Use a scale factor of 2.

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**9.1 Translating Figures and Using Vectors**

Translation Theorem: A translation is an isometry. Isometry- a congruence transformation Preimage- original figure Image- new figure

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**Write a rule for the translation of ABC to. A′B′C′**

Write a rule for the translation of ABC to A′B′C′. Then verify that the transformation is an isometry.

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**Name the vector and write its component form.**

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**The vertices of ∆LMN are L(2, 2), M(5, 3), and N(9, 1)**

The vertices of ∆LMN are L(2, 2), M(5, 3), and N(9, 1). Translate ∆LMN using the vector –2, 6.

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**A boat heads out from point A on one island toward point D on another**

A boat heads out from point A on one island toward point D on another. The boat encounters a storm at B, 12 miles east and 4 miles north of its starting point. The storm pushes the boat off course to point C, as shown. Write the component form of AB, BC, and CD.

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**9.2 Using Properties of Matrices**

Matrix- a rectangular arrangement of numbers in rows and columns Element- each number in the matrix Dimensions- row x column

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**9.3 Performing Reflections**

A reflection in a line (m) maps every point (P) in the plane to a point (P`) so that for each point, one of the following is true:

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Rules for Reflections If (a,b) is reflected in the x-axis, its image is (a,-b). If (a,b) is reflected in the y-axis, its image is (-a,b). If (a,b) is reflected in the line y = x, its image is (b,a). If (a,b) is reflected in the line y = -x, its image is (-b,-a).

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Examples

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**You and a friend are meeting on the beach shoreline**

You and a friend are meeting on the beach shoreline. Where should you meet to minimize the distance you must both walk?

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**Find the reflection of PQR in the x- axis using in matrix multiplication.**

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**9.4 Performing Rotations A rotation is an isometry**

Center of rotation- a fixed point in which a figure is turned about Angle of Rotation- the angle formed from rays drawn from the center of rotation to a point and its image

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Rules for Rotations These rules apply for counterclockwise rotations about the origin a 90o rotation (a,b) (-b,a) a 180o rotation (a,b) (-a,-b) a 270o rotation (a,b) (b,-a)

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Examples

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**9.5 Applying Compositions of Transformations**

Composition of Transformation- 2 or more transformations are combined to form a single transformation The composition of 2 (or more) isometries is an isometry.

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**Glide Reflection Example**

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Example

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**Reflections in Parallel Lines Theorem**

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Example

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**Reflection in Intersecting Lines Theorem**

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