**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.

**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 ABCare 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 ABand CD.Tell whether CDis a rotation of ABabout 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 PQRis 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 quadrilateralABCDwith verticesA(2, 1), B(4, 1), C(4, – 1), andD(1, – 1). Use a scale factor of2.

**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 ofABCto ** A′B′C′.Then verify that the transformation is an isometry.

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

**The vertices of ∆LMNare L(2, 2), M(5, 3), and N(9, 1).** Translate ∆LMNusing 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.

**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. 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. • P(-3,6) Q(-5,3) R(-1,2)

**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 180orotation (a,b) (-a,-b) • a 270orotation (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**