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Understanding Dilations in Ratios and Proportions

Learn how to create and identify dilations of plane figures using scale factors and the center of dilation. Practice solving dilation problems.

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Understanding Dilations in Ratios and Proportions

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  1. Ch. 7 Learning Goal: Ratios & Proportions • Learn to find equivalent ratios to create proportions (7-1) • Learn to work with rates and ratios (7-2) • Learn to use one or more conversion factors to solve rate problems (7-3) • Learn to solve proportions (7-4) • Learn to identify and create dilations of plane figures (7-5) • Learn to determine whether figures are similar, to use scale factors, and to find missing dimensions similar figures (7-6) • Learn to make comparisons between and find dimensions of scale drawings and actual objects (7-7) • Learn to make comparisons between and find dimensions of scale models and actual objects (7-8) • Learn to make scale models of solid figures (7-9)

  2. Pre-Algebra Homework Page 364 #7-12 & #20-24 (Spiral Review)

  3. 7-5 Dilations Warm Up Problem of the Day Lesson Presentation Pre-Algebra

  4. 7-5 Dilations 3 4 3 4 3 4 3 4 Pre-Algebra Warm Up Multiply. 1. 4  2. 12  3. 24  4. –36  9 3 18 –27 10 30 5. 4 2.5 6. 12 2.5

  5. Problem of the Day Every day, a plant grows to three times its size. Every night, it shrinks to half its size. After three days and nights, it is 6.75 in. tall. How tall was the plant at the start? 2 in.

  6. Today’s Learning Goal Assignment Learn to identify and create dilations of plane figures.

  7. Vocabulary dilation scale factor center of dilation

  8. Your pupils are the black areas in the center of your eyes. When you go to the eye doctor, the doctor may dilate your pupils, which makes them larger.

  9. Translations, reflections, and rotations are transformations that do not change the size or shape of a figure. A dilation is a transformation that changes the size, but not the shape, of a figure. A dilation can enlarge or reduce a figure.

  10. A scale factor describes how much a figure is enlarged or reduced. A scale factor can be expressed as a decimal, fraction, or percent. A 10% increase is a scale factor of 1.1, and a 10% decrease is a scale factor of 0.9.

  11. Helpful Hint A scale factor between 0 and 1 reduces a figure. A scale factor greater than 1 enlarges it.

  12. Additional Example 1A & 1B: Identifying Dilations Tell whether each transformation is a dilation. A. B. The transformation is not a dilation. The figure is distorted. The transformation is a dilation.

  13. A' A A' B' B B' C C' C' A B C Try This: Example 1A & 1B Tell whether each transformation is a dilation. A. B. The transformation is not a dilation. The figure is distorted. The transformation is a dilation.

  14. Additional Example 1C & 1D: Identifying Dilations Tell whether each transformation is a dilation. D. C. The transformation is not a dilation. The figure is distorted. The transformation is a dilation.

  15. A A A' A' B C C' B' B C C' B' Try This: Example 1C & 1D Tell whether each transformation is a dilation. D. C. The transformation is not a dilation. The figure is distorted. The transformation is a dilation.

  16. Every dilation has a fixed point that is the center of dilation. To find the center of dilation, draw a line that connects each pair of corresponding vertices. The lines intersect at one point. This point is the center of dilation.

  17. Additional Example 2: Dilating a Figure Dilate the figure by a scale factor of 1.5 with P as the center of dilation. Multiply each side by 1.5.

  18. G 1 cm 1 cm 2 cm 2 cm F’ H’ 1 cm F H 2 cm Try This: Example 2 Dilate the figure by a scale factor of 0.5 with G as the center of dilation. G 2 cm 2 cm F H 2 cm Multiply each side by 0.5.

  19. A’B’C’ A(4, 8) A’(4  2, 8  2) A’(8, 16) B(3, 2) B’(3  2, 2  2) B’(6, 4) C(5, 2) C’(5  2, 2  2) C’(10, 4) ABC Additional Example 3A: Using the Origin as the Center of Dilation Dilate the figure in Example 3A on page 363 by a scale factor of 2. What are the vertices of the image? Multiply the coordinates by 2 to find the vertices of the image. The vertices of the image are A’(8, 16), B’(6, 4), and C’(10, 4).

  20. Try This: Example 3A Dilate the figure by a scale factor of 2. What are the vertices of the image? 10 8 6 C 4 2 A B 10 8 4 6 2 0

  21. A’B’C’ A(2, 2) A’(2  2, 2  2) A’(4, 4) B(4, 2) B’(4  2, 2  2) B’(8, 4) C(2, 4) C’(2  2, 4  2) C’(4, 8) ABC Try This: Example 3A Continued The vertices of the image are A’(4, 4), B’(8, 4), and C’(4, 8).

  22. C’ B’ A’ Try This: Example 3A Continued 10 8 6 C 4 2 B A 10 8 4 6 2 0

  23. Dilate the figure in Example 3B by a scale factor of . What are the vertices of the image? 1 3 Multiply the coordinates by to find the vertices of the image. A’B’C’ C(6, 3) C’(6  , 3 ) C’(2, 1) B(9, 6) B’(9  , 6 ) B’(3, 2) A(3, 9) A’(3  , 9 ) A’(1, 3) ABC 1 3 1 3 1 3 1 3 1 3 1 3 1 3 Additional Example 3B: Using the Origin as the Center of Dilation The vertices of the image are A’(1, 3), B’(3, 2), and C’(2, 1).

  24. Try This: Example 3B Dilate the figure by a scale factor of 0.5. What are the vertices of the image? 10 C 8 6 B A 4 2 10 8 4 6 2 0

  25. A(4, 5) A’(4  0.5, 5  0.5) A’(2, 2.5) B(8, 5) B’(8  0.5, 5  0.5) B’(4, 2.5) A’B’C’ ABC C(4, 9) C’(4  0.5, 9  0.5) C’(2, 4.5) Try This: Example 3B Continued The vertices of the image are A’(2, 2.5), B’(4, 2.5), and C’(2, 4.5).

  26. C’ B’ A’ Try This: Example 3B Continued 10 C 8 6 B A 4 2 10 8 4 6 2 0

  27. 1. Tell whether the transformation is a dilation. A(0, 4) B(5,5) C(3,3) A’(0, 8) B’(10, 10) C’(6, 6) P C’ C A B B’ A’ Lesson Quiz yes 2 4 6 2. Dilate the figure by a scale factor of 1.5 with P as the center of dilation. -2 3. Dilate the figure by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of the image? A(2,4) B(5,6) C(6,1) -4 -6 A’(4,8) B’(10,12) C’(12,2)

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