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Understanding Similar Polygons: Proportions, Scale Factors, and Missing Measures

Learn about similar polygons, how to use a similarity statement, identify similar shapes, find missing lengths, and apply scale factors. Explore real-world examples and the relationship between corresponding angles and sides.

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Understanding Similar Polygons: Proportions, Scale Factors, and Missing Measures

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  1. LESSON 7–2 Similar Polygons

  2. Five-Minute Check (over Lesson 7–1) TEKS Then/Now New Vocabulary Key Concept: Similar Polygons Example 1: Use a Similarity Statement Example 2: Real-World Example: Identify Similar Polygons Example 3: Use Similar Figures to Find Missing Measures Theorem 7.1: Perimeter of Similar Polygons Example 4: Use a Scale Factor to Find Perimeter Lesson Menu

  3. There are 480 sophomores and 520 juniors in a high school. Find the ratio of juniors to sophomores. A. 10:8 B. 13:12 C. 19:17 D. 22:20 5-Minute Check 1

  4. A strip of wood molding that is 33 inches long is cut into two pieces whose lengths are in the ratio of 7:4. What are the lengths of the two pieces? A. 7 in., 4 in. B. 14 in., 8 in. C. 18 in., 15 in. D. 21 in., 12 in. 5-Minute Check 2

  5. A. 7 B. 8 C. 9 D. 10 5-Minute Check 3

  6. A. 2.75 B. 3.25 C. 3.75 D. 4.25 5-Minute Check 4

  7. A. 4 B. 3 C. 2 D. 1 5-Minute Check 5

  8. The standard ratio of a photo’s width to its length is . What is the length of a photo that has a width of 14 inches? A. 9.3 inches B. 17 inches C. 20 inches D. 56 inches 5-Minute Check 6

  9. Targeted TEKS G.10(B) Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change. Mathematical Processes G.1(F), G.1(G) TEKS

  10. You used proportions to solve problems. • Use proportions to identify similar polygons. • Solve problems using the properties of similar polygons. Then/Now

  11. similar polygons • scale factor Vocabulary

  12. Concept

  13. Use a Similarity Statement If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides. Example 1

  14. Use a Similarity Statement Use the similarity statement. ΔABC ~ ΔRST Answer: Congruent Angles: A R, B S,C T Example 1

  15. A.HGK  QPR B. C.K  R D.GHK  QPR If ΔGHK ~ ΔPQR, determine which of the following similarity statements is not true. Example 1

  16. Identify Similar Polygons A. MENUSTan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu: Example 2

  17. Identify Similar Polygons Step 1 Compare corresponding angles. Since all angles of a rectangle are right angles and right angles are congruent, corresponding angles are congruent. Step 2 Compare corresponding sides. Answer:Since corresponding sides are not proportional, ABCD is not similar to FGHK. So, the menus are not similar. Example 2

  18. Identify Similar Polygons B. MENUSTan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu: Example 2

  19. Identify Similar Polygons Step 1 Compare corresponding angles. Since all angles of a rectangle are right angles and right angles are congruent, corresponding angles are congruent. Step 2 Compare corresponding sides. Example 2

  20. Answer:Since corresponding sides are proportional, ABCD ~ RSTU. So, the menus are similar with a scale factor of . 4 __ 5 Identify Similar Polygons Example 2

  21. A.BCDE ~ FGHI, scale factor = B.BCDE ~ FGHI, scale factor = C.BCDE ~ FGHI, scale factor = D.BCDE is not similar to FGHI. 1 4 3 __ __ __ 2 5 8 Original: New: A. Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. Example 2

  22. A.BCDE ~ WXYZ, scale factor = B.BCDE ~ WXYZ, scale factor = C.BCDE ~ WXYZ, scale factor = D.BCDE is not similar to WXYZ. 1 4 3 __ __ __ 2 5 8 Original: New: B. Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. Example 2

  23. Use Similar Figures to Find Missing Measures A.The two polygons are similar. Find x. Use the congruent angles to write the corresponding vertices in order. polygon ABCDE ~ polygon RSTUV Example 3

  24. 9 __ 2 Answer: x = Use Similar Figures to Find Missing Measures Write a proportion to find x. Similarity proportion Cross Products Property Multiply. Divide each side by 4. Simplify. Example 3

  25. Use Similar Figures to Find Missing Measures B.The two polygons are similar. Find y. Use the congruent angles to write the corresponding vertices in order. polygon ABCDE ~ polygon RSTUV Example 3

  26. 13 __ 3 Answer: y = Use Similar Figures to Find Missing Measures Similarity proportion AB = 6, RS = 4, DE = 8, UV = y + 1 Cross Products Property Multiply. Subtract 6 from each side. Divide each side by 6 and simplify. Example 3

  27. A. The two polygons are similar. Solve for a. A.a = 1.4 B.a = 3.75 C.a = 2.4 D.a = 2 Example 3

  28. B. The two polygons are similar. Solve for b. A. 1.2 B. 2.1 C. 7.2 D. 9.3 Example 3

  29. Concept

  30. Use a Scale Factor to Find Perimeter If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon. Example 4

  31. The scale factor ABCDE to RSTUV is or . Write a proportion to find the length of DC. 4 __ AE ___ 7 VU Since DC AB and AE  DE, the perimeter of ABCDE is 6 + 6 + 6 + 4 + 4 or 26. Use a Scale Factor to Find Perimeter Write a proportion. 4(10.5) = 7 ● DC Cross Products Property 6 = DC Divide each side by 7. Example 4

  32. Use a Scale Factor to Find Perimeter Use the perimeter of ABCDE and scale factor to write a proportion. Let x represent the perimeter of RSTUV. Theorem 7.1 Substitution 4x = (26)(7) Cross Products Property x = 45.5 Solve. Example 4

  33. Use a Scale Factor to Find Perimeter Answer:The perimeter of ABCDE is 26 and the perimeter of RSTUV is 45.5. Example 4

  34. If LMNOP ~ VWXYZ, find the perimeter of each polygon. A.LMNOP = 40, VWXYZ = 30 B.LMNOP = 32, VWXYZ = 24 C.LMNOP = 45, VWXYZ = 40 D.LMNOP = 60, VWXYZ = 45 Example 4

  35. LESSON 7–2 Similar Polygons

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