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Exploring Dilations: Transformations in Geometry

Learn about dilations in the coordinate plane and real-world applications. Understand scale factors and symmetry. Practice drawing dilations and verify them as similarity transformations.

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Exploring Dilations: Transformations in Geometry

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 9–5) CCSS Then/Now Key Concept: Dilation Example 1: Draw a Dilation Example 2: Real-World Example: Find the Scale of a Dilation Key Concept: Dilations in the Coordinate Plane Example 3: Dilations in the Coordinate Plane Lesson Menu

  3. State whether the figure appears to have line symmetry. If so, how many lines of symmetry does it have? A. yes; 4 lines B. yes; 3 lines C. yes; 2 lines D. This figure does not have line symmetry. 5-Minute Check 1

  4. State whether the figure appears to have line symmetry. If so, how many lines of symmetry does it have? • A. yes; 8 lines • B. yes; 4 lines • yes; 2 lines • This figure does not line symmetry. 5-Minute Check 2

  5. The figure has rotational symmetry. State the order and magnitude of symmetry. A. 5; 72° B. 5; 45° C. 6; 60° D. 6; 72° 5-Minute Check 3

  6. The figure has rotational symmetry. State the order and magnitude of symmetry. A. 8; 60° B. 8; 45° C. 10; 45° D. 10; 36° 5-Minute Check 4

  7. What is the order and magnitude of symmetry of a regular hexagon? A. order 2, magnitude 180° B. order 3, magnitude 120° C. order 6, magnitude 60° D. order 12, magnitude 30° 5-Minute Check 5

  8. Content Standards G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.SRT.1 Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Mathematical Practices 1 Make sense of problems and persevere in solving them. 5 Use appropriate tools strategically. CCSS

  9. You identified dilations and verified them as similarity transformations. • Draw dilations. • Draw dilations in the coordinate plane. Then/Now

  10. Concept

  11. Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Since k > 1, the dilation is an enlargement of trapezoid PQRS. Example 1

  12. Locate P', Q', R', and S' so that Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Example 1

  13. Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Answer: Draw trapezoid P'Q'R'S'. Example 1

  14. Which diagram shows the dilation image of ΔLMN with center C and ? A. B. C.D. Example 1

  15. Find the Scale Factor of a Dilation PUPPETS To create the illusion of a “life-sized” image, puppeteers sometimes use a light source to show an enlarged image of a puppet projected on a screen or wall. Suppose that the distance between a light source L and the puppet is 24 inches (LP). To what distance PP' should you place the puppet from the screen to create a 49.5-inch tall shadow (I'M') from a 9-inch puppet? Example 2

  16. Find the Scale Factor of a Dilation Understand This problem involves a dilation. The center of the dilation is L, LP = 24 in., IM = 9 in., I'M' = 49.5 in. You are asked to find PP'. Plan Find the scale factor of the dilation from the preimage IM to the image I'M'. Use the scale factor to find LP and then use LP and LP' to find PP'. Example 2

  17. Find the Scale Factor of a Dilation Solve The scale factor k of the enlargement is the ratio of the length on the image to a corresponding length on the preimage. Scale factor of image image = I'M', preimage = IM Divide. Example 2

  18. Find the Scale Factor of a Dilation Use this scale factor of 5.5 to find LP'. LP'= k(LP) Definition of dilation = 5.5(24)k = 5.5 and LP = 24 = 132 Multiply. Use LP' and LP to find PP'. LP + PP' = LP'Segment Addition 24 + PP'= 132LP = 24 and LP' = 132 PP' = 108 Subtract 24 from each side. Example 2

  19. Find the Scale Factor of a Dilation Answer: So, the puppet should be placed so that the distance from it to the screen (PP') is 108 inches. CheckSince the dilation is an enlargement, the scale factor should be greater than 1. Since 5.5 > 1, the scale factor is reasonable. Example 2

  20. PUPPETS Suppose you have a similar situation with the puppet and light source. The distance between the light source L and the puppet is 30 inches (LP). To what distance should you place the puppet from the screen to create a 54-inch tall shadow (I'M') from a 6-inch puppet? A. 100 inches B. 180 inches C. 220 inches D. 240 inches Example 2

  21. Concept

  22. Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Graph the image of EFGH after a dilation centered at the origin with a scale factor of Multiply the x- and y-coordinates of each vertex by the scale factor, Dilations in the Coordinate Plane Example 3

  23. Dilations in the Coordinate Plane Graph the preimage and image. Answer:E'(–2, 1), F'(–1, 2), G'(2, 1), H'(–1, –2) Example 3

  24. A. B. C.D. none of the above Triangle ABChas vertices A(–1, 1), B(2, –2), and C(–1, –2). Find the image of ΔABC after a dilation centered at the origin with a scale factor of 2. Sketch the preimage and the image. Example 3

  25. End of the Lesson

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