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Dilations

Dilations. Lesson 14.5 Pre-AP Geometry. Lesson Focus. There are transformations that do not preserve distance. This lesson introduces one such transformation, called a dilation. Basic Terms. Dilation

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Dilations

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  1. Dilations Lesson 14.5 Pre-AP Geometry

  2. Lesson Focus There are transformations that do not preserve distance. This lesson introduces one such transformation, called a dilation.

  3. Basic Terms Dilation A transformation that changes the size of a figure by a scale factor to create a similar figure. A dilation is not rigid. DO,k A dilation with a center O and a nonzero scale factor k maps any point P to a point P’. 1) If k > 0, P’ lies on and OP’ = k · OP. 2) If k < 0, P’ lies on the ray opposite and OP’ = |k| · OP.

  4. Basic Terms Expansion A dilation where the scale factor |k| > 1. Contraction A dilation where the scale factor |k| < 1.

  5. Basic Terms Similarity Mapping A transformation that maps any geometric figure to a similar geometric figure. A dilation is not an isometry as distances are not preserved.

  6. Theorem 14-5 A dilation maps a triangle to a similar triangle.

  7. Corollary 1 A dilation maps an angle to a congruent angle.

  8. Corollary 2 A dilation DO,k maps any segment to a parallel segment |k| times as long.

  9. Corollary 3 A dilation DO,k maps any polygon to a similar polygon whose area is k2 times as large.

  10. Practice • Given: A(3, 6), B(-3, -3), and C(-6, 0). Find: (a) DO,2 ; (b) DO, -1/3 2. A dilation with the origin, O, as center maps (3, 4) to (9, 12). Find the scale factor. Is the dilation an expansion or a contraction? • A dilation with the origin, O, as center maps (-3, 4) → (1, -4/3). Find the scale factor. Is the dilation an expansion or contraction?

  11. Review • Which transformations are isometries? • If g(x) = 7 – 2x, find the image of 3 and the preimage of -5. • If R: (x, y) → (x – 2, y + 3), find the image of (-3, 1). • Find the image of (-1, 4) when reflected in each line. a. the x-axis b. the y-axis c. the line y = x

  12. Written Exercises Problem Set 14.5, p. 596: # 2 – 8 (even) Handout: 14-5

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