Chapter 9Transformations

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

4.8 Transformations

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

Congruence transformations

moves every point of a figure the same distance in the same direction.

Coordinate notation: (x , y) (x + a, y + b)

A Translation

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?

Example

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)

A Reflection

Reflect a figure in the x-axis

Example

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

Rotation

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)

Examples

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.

a. b.

Name the type of transformation shown.

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.

6.7 Dilations

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

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.

Examples

Translation Theorem: A translation is an isometry.

Isometry- a congruence transformation Preimage- original figure Image- new figure

9.1 Translating Figures and Using Vectors

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.

a.

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. 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.

Matrix- a rectangular arrangement of numbers in rows and columns

Element- each number in the matrix Dimensions- row x column

9.2 Using Properties of Matrices

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:

9.3 Performing Reflections

m m

P̀

P

P`

P

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).

Rules for Reflections

Examples

6

4

2

-2

-4

-6

-10 -5 5 10

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

6

4

2

-2

-4

-6

-10 -5 5 10

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

P(-3,6) Q(-5,3) R(-1,2)

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

9.4 Performing 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)

Rules for Rotations

Examples

6

4

2

-2

-4

-6

-10 -5 5 10

Composition of Transformation- 2 or more transformations are combined to form a single transformation

The composition of 2 (or more) isometries is an isometry.

9.5 Applying Compositions of Transformations

Glide Reflection Example6

4

2

-2

-4

-6

-10 -5 5 10

Example6

4

2

-2

-4

-6

-10 -5 5 10

Reflections in Parallel Lines Theorem

mk

Example

mk

Reflection in Intersecting Lines Theorem

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