Chapter 9

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## Chapter 9

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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.**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.**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).**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)**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.