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Understanding Dilations and Scale Factors

Learn how to identify dilations, determine scale factors, and verify similarity after a dilation. Explore real-world examples and practice problems.

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Understanding Dilations and Scale Factors

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 7–5) NGSSS Then/Now New Vocabulary Key Concept: Types of Dilations Example 1: Identify a Dilation and Find Its Scale Factor Example 2: Real-World Example: Find and Use a Scale Factor Example 3: Verify Similarity after a Dilation Lesson Menu

  3. A B C D Find the value of a. A. 1 B. 2 C. 3.5 D. 5 5-Minute Check 1

  4. A B C D Find the value of n. A. 45 B. 54 C. 67 D. 76 5-Minute Check 2

  5. A B C D Find the value of x. A. 8.5 B. 9 C. 10 D. 11 5-Minute Check 3

  6. A B C D Find the value of x. A. 9 B. 10 C. 11 D. 12 5-Minute Check 4

  7. A B C D Find the value of x. A. 1 B. 2.4 C. 2.6 D. 4.5 5-Minute Check 5

  8. MA.912.G.2.4Apply transformations to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. MA.912.G.2.6 Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane. NGSSS

  9. You identified congruence transformations. (Lesson 4–7) • Identify similarity transformations. • Verify similarity after a similarity transformation. Then/Now

  10. dilation • similarity transformation • center of dilation • scale factor of a dilation • enlargement • reduction Vocabulary

  11. Concept

  12. Identify a Dilation and Find Its Scale Factor A. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. B is smaller than A, so the dilation is a reduction. Example 1

  13. 1 2 __ __ 2 4 Answer: So the scale factor is or . Identify a Dilation and Find Its Scale Factor The distance between the vertices at (2, 2) and (2, –2) for A is 4 and from the vertices at (1, 1) and (1, –1) for Bis 2. Example 1

  14. Identify a Dilation and Find Its Scale Factor B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. B is larger than A, so the dilation is an enlargement. Example 1

  15. 6 __ Answer: So the scale factor is or 3. 2 Identify a Dilation and Find Its Scale Factor The distance between the vertices at (3, 3) and (–3, 3) for A is 6 and from the vertices at (1, 1) and (–1, 1) for Bis 2. Example 1

  16. A B C D A. reduction; B. reduction; C. enlargement; 2 D. enlargement; 3 1 1 __ __ 3 2 A. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. Example 1

  17. A B C D A. reduction; B. reduction; C. enlargement; D. enlargement; 2 1 2 3 __ __ __ 3 3 2 B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. Example 1

  18. Find and Use a Scale Factor PHOTOCOPYING A photocopy of a receipt is 1.5 inches wide and 4 inches long. By what percent should the receipt be enlarged so that its image is 2 times the original? What will be the dimensions of the enlarged image? To enlarge the receipt 2 times the original, use a scale factor of 2. Written as a percent, the scale factor is (2 ● 100%) or 200%. Now, find the dimensions of the enlarged receipt. width: 1.5 in. ● 200% = 3 in. length: 4 in. ● 200% = 8 in. Answer: The enlarged receipt will be 3 inches by 8 inches. Example 2

  19. A B C D PHOTOGRAPHS Mariano wants to enlarge a picture he took that is 4 inches by 7.5 inches. He wants it to fit perfectly into a frame that is 400% of the original size. What will be the dimensions of the enlarged photo? A. 15 inches by 25 inches B. 8 inches by 15 inches C. 12 inches by 22.5 inches D. 16 inches by 30 inches Example 2

  20. Verify Similarity after a Dilation A. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: M(–6, –3), N(6, –3), O(–6, 6) image: D(–2, –1), F(2, –1), G(–2, 2) Graph each figure. Since M and D are both right angles, M  D. Show that the lengths of the sides that include M and D are proportional. Example 3

  21. Verify Similarity after a Dilation Use the coordinate grid to find the lengths of the vertical segments MO and DG and the horizontal segments MN and DF. Answer:Since the lengths of the sides that include M and D are proportional, ΔMNO ~ ΔDFG by SAS Similarity. Example 3

  22. Verify Similarity after a Dilation B. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: G(2, 1), H(4, 1), I(2, 0), J(4, 0) image: Q(4, 2), R(8, 2), S(4, 0), T(8, 0) Example 3

  23. Verify Similarity after a Dilation Use the Distance Formula to find the length of each side. Example 3

  24. Answer: Verify Similarity after a Dilation Use the Distance Formula to find the length of each side. Find and compare the ratios of corresponding sides. Example 3

  25. A B C D A. B. C. D. 1 1 3 3 __ __ __ __ 2 3 2 4 A. Graph the original figure and its dilated image. Then determine the scale factor of the dilation.original: B(–7, –2), A(5, –2), D(–7, 7)image: J(–3, 0), K(1, 0), L(–3, 3) Example 3

  26. A B C D A.2 B. C.3 D.4 1 __ 3 B. Graph the original figure and its dilated image. Then determine the scale factor of the dilation.original: A(4, 3), B(6, 3), C(4, 2), D(6, 2)image: E(6, 4), F(10, 4), G(6, 2), H(10, 2) Example 3

  27. End of the Lesson

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