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Ratios, Extended Ratios, and Proportions in Geometry

Learn how to write and simplify ratios, use extended ratios, and solve proportions in parallelograms and trapezoids. Apply geometric methods to solve problems.

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Ratios, Extended Ratios, and Proportions in Geometry

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 6) CCSS Then/Now New Vocabulary Example 1: Real-World Example: Write and Simplify Ratios Example 2: Use Extended Ratios Key Concept: Cross Products Property Example 3: Use Cross Products to Solve Proportions Example 4: Real-World Example: Use Proportions to Make Predictions Key Concept: Equivalent Proportions Lesson Menu

  3. Complete the statement about parallelogram LMNO. ? A. B. C. D. 5-Minute Check 1

  4. Complete the statement about parallelogram LMNO. ? A. B. C. D. 5-Minute Check 2

  5. Complete the statement about parallelogram LMNO.OLM  ____ ? A. MNO B. LON C. MPN D. MPL 5-Minute Check 3

  6. Complete the statement about parallelogram LMNO. ? A. B. C. D. 5-Minute Check 4

  7. Find the measure of each interior angle. • mA = 60, mB = 120, mC = 60, mD = 120 • B.mA = 65, mB = 115, mC = 65, mD = 115 • C.mA = 65, mB = 115, mC = 60, mD = 120 • D.mA = 70, mB = 110, mC = 70, mD = 110 5-Minute Check 5

  8. Which of the statements about an isosceles trapezoid is false? A. The diagonals are congruent. B. Two sides are both congruent and parallel. C. Two pairs of base angles are congruent. D. One pair of sides are congruent. 5-Minute Check 6

  9. Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning. CCSS

  10. You solved problems by writing and solving equations. • Write ratios. • Write and solve proportions. Then/Now

  11. ratio • extended ratios • proportion • extremes • means • cross products Vocabulary

  12. 0.3 can be written as Write and Simplify Ratios SCHOOL The number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth. To find this ratio, divide the number of athletes by the total number of students. Answer: The athlete-to-student ratio is 0.3. Example 1

  13. The country with the longest school year is China, with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.) A. 0.3 B. 0.5 C. 0.7 D. 0.8 Example 1

  14. Just as the ratio or 5:12 is equivalent to or 5x:12x, the extended ratio 5:12:13 can be written as 5x:12x:13x. Write and solve an equation to find the value of x. 5 ___ 12 5x ______ 12x Use Extended Ratios In ΔEFG, the ratio of the measures of the angles is 5:12:13. Find the measures of the angles. Example 2

  15. Use Extended Ratios 5x + 12x + 13x = 180 Triangle Sum Theorem 30x = 180 Combine like terms. x = 6 Divide each side by 30. Answer: So, the measures of the angles are 5(6) or 30, 12(6) or 72, and 13(6) or 78. Example 2

  16. The ratios of the angles in ΔABC is 3:5:7. Find the measure of the angles. A. 30, 50, 70 B. 36, 60, 84 C. 45, 60, 75 D. 54, 90, 126 Example 2

  17. Concept

  18. Use Cross Products to Solve Proportions A. Original proportion Cross Products Property Multiply. Divide each side by 6. y = 27.3 Answer:y = 27.3 Example 3

  19. Use Cross Products to Solve Proportions B. Original proportion Cross Products Property Simplify. Add 30 to each side. Divide each side by 24. Answer:x = –2 Example 3

  20. A. A.b = 0.65 B.b = 4.5 C.b = –14.5 D.b = 147 Example 3

  21. B. A.n = 9 B.n = 8.9 C.n = 3 D.n = 1.8 Example 3

  22. Use Proportions to Make Predictions PETS Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique’s school, predict the total number of students with a dog or a cat. Write and solve a proportion that compares the number of students who have a pet to the number of students in the school. Example 4

  23. Use Proportions to Make Predictions ← Students who have a pet ← total number of students 18 ● 870 = 30x Cross Products Property 15,660 = 30x Simplify. 522 = x Divide each side by 30. Answer: Based on Monique's survey, about 522 students at her school have a dog or a cat for a pet. Example 4

  24. Brittany randomly surveyed 50 students and found that 20 had a part-time job. If there are 810 students in Brittany's school, predict the total number of students with a part-time job. A. 324 students B. 344 students C. 405 students D. 486 students Example 4

  25. Concept

  26. End of the Lesson

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