2 4.8 TransformationsAn operation that moves or changes a geometric figure (a preimage) in some way to produce a new figure (an image).
3 Congruence transformations changes the position of the figure without changing the size or shape.TranslationReflectionRotation
4 A Translationmoves every point of a figure the same distance in the same direction.Coordinate notation: (x , y) (x + a, y + b)
5 ExampleThe 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?
6 A ReflectionUses 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)
13 6.7 DilationsA 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.
14 The scale factor of a dilation is the ratio of the side length of the image to the corresponding side length of the original figureCoordinate notation for a dilation with respect to the origin: (x ,y) ( kx, ky)Reduction: 0 < k < 1Enlargement : k > 1
15 ExamplesDraw 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.
20 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.
21 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.
22 9.2 Using Properties of Matrices Matrix- a rectangular arrangement of numbers in rows and columnsElement- each number in the matrixDimensions- row x column
23 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:
24 Rules for ReflectionsIf (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).
26 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?
27 Find the reflection of PQR in the x- axis using in matrix multiplication.
28 9.4 Performing Rotations A rotation is an isometry Center of rotation- a fixed point in which a figure is turned aboutAngle of Rotation- the angle formed from rays drawn from the center of rotation to a point and its image
29 Rules for RotationsThese rules apply for counterclockwise rotations about the origina 90o rotation (a,b) (-b,a)a 180o rotation (a,b) (-a,-b)a 270o rotation (a,b) (b,-a)
31 9.5 Applying Compositions of Transformations Composition of Transformation- 2 or more transformations are combined to form a single transformationThe composition of 2 (or more) isometries is an isometry.