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8.1: Dilations and Scale Factors

8.1: Dilations and Scale Factors. Expectation: G3.2.1: Know the definition of dilation and find the image of a figure under a given dilation. Rigid Transformations.

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8.1: Dilations and Scale Factors

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  1. 8.1: Dilations and Scale Factors Expectation: G3.2.1: Know the definition of dilation and find the image of a figure under a given dilation. 8.1 Dilations and Scale Factor

  2. Rigid Transformations • Remember, a rigid transformation preserves size and shape. In other words, the preimage and image are congruent. Not all transformations are congruent. 8.1 Dilations and Scale Factor

  3. Dilations • A dilation is a type of transformation that is not rigid. • Dilations preserve shape, but not necessarily size. Because the size can change, dilations are sometimes called size changes. 8.1 Dilations and Scale Factor

  4. If you did 2 dimensional size changes in Algebra, then you have already worked with dilations. 8.1 Dilations and Scale Factor

  5. Dilations • Defn: A transformation is a dilation iff every point P has an image P’ such that PP’ passes through a point, O, such that OP’ = k  OP. O is called the center of the dilation and k is the magnitude (scale factor) of the size change. 8.1 Dilations and Scale Factor

  6. Dilations • In general, the dilation of a point P(x,y) with scale factor k can be found using the following rule: • P’ = (kx,ky) • Another way to write this is to use D to represent a dilation. Then D(x,y) = (kx,ky). 8.1 Dilations and Scale Factor

  7. If D(x,y) = (3x,3y), what is the image of the point (-5,8)? • What is the scale factor of the dilation? 8.1 Dilations and Scale Factor

  8. 3 Types of Dilations • If |k| < 1, the dilation is a contraction • If |k| > 1, then dilation is an expansion • If |k| = 1, the dilation is an identity dilation 8.1 Dilations and Scale Factor

  9. Let O(0,0) and A(4,6). What is OA? 8.1 Dilations and Scale Factor

  10. How is OA’ related to OA? • The distance from the origin to the image of a point transformed by a dilation with scale factor k is _____ times the distance from the origin to the preimage. 8.1 Dilations and Scale Factor

  11. Let A(2,6) and B(-2,-4). What is AB? What is the slope of AB? 8.1 Dilations and Scale Factor

  12. Size Change Distance Theorem • The image of a segment transformed by a dilation with scale factor k is _________ to and ____ times the length of the preimage. 8.1 Dilations and Scale Factor

  13. Size Changes Without Coordinates • 1. Draw a triangle. Label the vertices A, B and C. • 2. Locate a point, O, the center of the dilation (it can be anywhere). • 3. Draw OA, OB and OC. • 4. Determine OA, OB and OC. 8.1 Dilations and Scale Factor

  14. 5. Let k = 2 and determine OA’, OB’ and OC’. • 6. Mark A’ on OA, B’ on OB and C’ on OC. • 7. Connect A’, B’ and C’ to form the dilation image of triangle ABC. 8.1 Dilations and Scale Factor

  15. Assignment • pages 502-505, • # 16, 20, 24, 25, 26, 28, 32, 35-38 (all) 8.1 Dilations and Scale Factor

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