 # Chapter 9 Transformations.

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Chapter 9 Transformations

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

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

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

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?

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)

Example Reflect a figure in the x-axis

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

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)

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.

Name the type of transformation demonstrated in each picture.

Name the type of transformation shown.

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.

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

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.

9.1 Translating Figures and Using Vectors
Translation Theorem: A translation is an isometry. Isometry- a congruence transformation Preimage- original figure Image- new figure

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.

Name the vector and write its component form.

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.

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.

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

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:

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).

Examples

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?

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

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

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)

Examples

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.

Glide Reflection Example

Example

Reflections in Parallel Lines Theorem

Example

Reflection in Intersecting Lines Theorem