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Warm-Up

Warm-Up. 1) Simplify each ratio. A) 12/15 B) 14/56 C) 21/6 D) 90/450 2) Find the perimeter. . 22. 11. 6. 20. 18. 7. 13. 10. 31. Real World Problem.

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Warm-Up

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  1. Warm-Up • 1) Simplify each ratio. • A) 12/15 B) 14/56 C) 21/6 D) 90/450 • 2) Find the perimeter. 22 11 6 20 18 7 13 10 31

  2. Real World Problem • On a package of microwavable meatballs, it states that it takes 1 minute to cook 6 meatballs. It also states that you need to add a minute for each additional 6 meatballs. You have 20 meatballs. How much time do you need to cook them?

  3. Simplifying ratios • Usually expressed in lowest terms: • : • Are expressed without units

  4. Different Units? • If there are different units, we convert so that they match (like in the previous example). • Or, we call them unit rates such as mpg, mph, etc…

  5. Using ratios • The triangles are similar. Find x. x 45 20 30

  6. Using Ratios • Perimeter of a rectangle is 60. Ratio of length to width is 3:2. Find the length and width.

  7. Real World Application: Construction • Video: http://player.discoveryeducation.com/index.cfm?guidAssetId=A5F37EB8-AC20-45FF-9878-F793A44BB6D1&blnFromSearch=1&productcode=US

  8. Properties with Ratios • Cross-Product Property • Reciprocal Property

  9. Proportions • Proportion: an equation that equates 2 ratios. • Use cross-products to solve.

  10. Warm-Up • #1-2 to turn in. • 1) What did you do in lieu of this class yesterday? If ACT, how was it? • 2) What is your favorite topic to discuss? Critical Thinking: • 3) What is the shortest distance between any two points?

  11. Continued Practice • Do textbook p. 461-462 #6-24, 34-44 evens

  12. Real World Application: Animals • Video: http://player.discoveryeducation.com/index.cfm?guidAssetId=F804E0DA-1249-40F2-8CF8-072A8E3A84E6&blnFromSearch=1&productcode=US

  13. Additional Properties of Proportions • If , then • If , then

  14. Using Properties of Proportions • Tell whether the statement is true.

  15. Proportion And Figures • When discussing proportion as it relates to figures, we state that those figures are similar • This means that all the angles are the same and the sides are proportional

  16. From One Figure to Another • If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. Ex: Scale factor of this triangle is 1:2 9 4.5 3 6

  17. Using Scale • Quadrilateral JKLM is similar to PQRS. Find the value of z. R K L S 15 Q z 6 P J M 10 15z = 60 z = 4

  18. Using Ratios • If AB:BC is 7:4, find m. B 19m + 3 12m A C

  19. Word Problems • Example: Imagine a 6 foot tall man. He comes across a race of people that are, on average, only 6 inches tall. Proportionally, how much tall is the man?

  20. Word Problems • Example 2: Babe Ruth had a 35 in. baseball bat. In Louisville, Kentucky, there is a giant replica of this bat. It stands 120 feet tall and its base has a diameter of 9 feet. • A) What is the height of the replica in inches? • B) What is the diameter of the base of the real bat?

  21. Additional Example?

  22. Practice • Worksheet

  23. Homework • Blue book p. 90, 92

  24. Warm-Up • 1) Find the scale factor for ∆MNQ → ∆M’N’Q’. Find the values of the variables. • Find the value of SU. M M’ x 18 Q’ 12 Q 20 y 15 N’ N P Q 8 10 R S 10 T U

  25. Homework ?s

  26. Continued Practice • Take 5-10 minutes to finish the worksheet from yesterday

  27. Identifying Similar Polygons Definition: Similar Polygons • Two polygons with congruent corresponding angles and corresponding sides that are proportional.

  28. Similar polygons • If ABCD ~ EFGH, then G H C D F B A E

  29. Similar polygons • Given ABCD ~ EFGH, solve for x. G H C D 6 x B F 2 4 A E 2x = 24 x = 12

  30. Additional Work with Similarity In the diagram . Find the length of BD. A 30 16 B C x 10 D E

  31. In the diagram • Solve for DE. A 5 2 D B 3 E C

  32. Theorem • If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. • If KLMN ~ PQRS, then

  33. Given ABC ~ DEF, find the scale factor of ABC to DEF and find the perimeter of each polygon. E P = 8 + 12 + 20 = 40 B P = 4 + 6 + 10 = 20 12 20 10 6 A C D F 8 4 CORRESPONDING SIDES 4 : 8 1 : 2

  34. Similarity and Word Problems • A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The titanic itself was 882.75ft long. How wide was it?

  35. Above is Bev Doolittle’s Music In the Wind. The actual painting is 12 inches high. Assume the image above is 1.25 in by 4.375 in. How wide is the original?

  36. Practice • P. 462 #45-47 • P. 463 #54-56 • P. 469 #23-30 • P. 468 #9-16

  37. Warm-Up Q K 8 P 105° • ∆KLM ~ ∆PQR • 1) Find the scale factor. • 2) Find the length of segment PQ. • 3) If the measures of the angles of a triangle sum to 180, find the measure of angle Q. 15 45° L M 10 3:2 12 30° R

  38. It’s Lean, It’s Green, It’s… • The geometric mean of two positive numbers a and b is the positive number x such that • Find the geometric mean of 8 and 18. • Solution: 12

  39. Geometric mean • Find the geometric mean of the two numbers. • 3 and 27 √(3)(27) = √81 = 9 • 4 and 16 √(4)(16) = √64 = 8 • 5 and 15 √(5)(15) = √75 = 5√3

  40. Another Way to Solve • The geometric mean of ‘a’ and ‘b’ is √ab • Using this method, find the geometric mean of 4 and 9 • Solution: 6, since √(4*9) = √36 = 6.

  41. Practice • P. 479 #4-6 • P. 611 #76-79

  42. Makeup Work • For the remainder of the class period, work on any assignments you need to make up or any tests you need to correct or retake.

  43. Warm-Up • Find the geometric mean of the two numbers. • 1) 3 and 27 2) 4 and 16 • 3) 7 and 28 4) 2 and 40 • 5) Solve. (2/3)n + 8 = (1/2)n + 2

  44. ?s

  45. Extended Ratio • Sometimes when working with objects, we are given an extended ratio of how items relate to each other. • Extended ratio—a ratio with three or more relationships

  46. Example • The measures of the angles in triangle JKL are in the extended ratio of 1:2:3. Find the measures of the angles.

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