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Lesson 7-1

Lesson 7-1. Geometric and Arithmetic Means. Objectives. Find the arithmetic mean between two numbers (their average) Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. Vocabulary.

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Lesson 7-1

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  1. Lesson 7-1 Geometric and Arithmetic Means

  2. Objectives • Find the arithmetic mean between two numbers (their average) • Find the geometric mean between two numbers • Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse

  3. Vocabulary • Arithmetic Mean – between two numbers is their average (add the two numbers and divide by 2) • Geometric Mean – between two numbers is the positive square root of their product

  4. √ (6•16) = √96 ≈ 9.8 √ (5•10) = √50 ≈ 7.07 √ (2•14) = √28 ≈ 5.29 √ (4•8) = √32 ≈ 5.66 Means Arithmetic Mean (AM): is the average of two numbers example: 4 and 10 AM = (4 + 10)/2 = 14/2 = 7 Geometric Mean (GM): is the square root of their product example: 4 and 10 GM = √(4•10) = √40 ≈ 6.32 Find the AM and GM of the following numbers AM GM a. 6 and 16 b. 4 and 8 c. 5 and 10 d. 2 and 14 (6+16)/2 = 11 (4+8)/2 = 6 (5+10)/2 = 7.5 (2+14)/2 = 8

  5. Example 1a Find the geometric mean between 2 and 50. Let x represent the geometric mean. Definition of geometric mean Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is 10.

  6. Example 1b a. Find the geometric mean between 3 and 12. b. Find the geometric mean between 4 and 20. Answer: 6 Answer: 8.9

  7. Application of Geometric Mean hypotenuse (length = a + b) a From similar triangles a x --- = --- x b altitude b x Geometric mean of two numbers a, b is square root of their product, ab The length of an altitude, x, from the 90° angle to the hypotenuse is the geometric mean of the divided hypotenuse x = ab

  8. x = √6 • 14 = √84 Example of Geometric Mean 6 altitude 14 x To find x, the altitude to the hypotenuse, we need to find the two pieces the hypotenuse has been divided into: a 6 piece and a 14 piece. The length of the altitude, x, is the geometric mean of the divided hypotenuse. ≈ 9.17

  9. Example 2a Cross products Take the positive square root of each side. Use a calculator. Answer: CD is about 12.7.

  10. Example 2b Answer: about 8.5

  11. Find c and d in is the altitude of right triangle JKL.Use Theorem 7.2 to write a proportion. Example 3 Cross products Divide each side by 5.

  12. is the leg of right triangle JKL. Use the Theorem 7.3 to write a proportion. Answer: Example 3 cont Cross products Take the square root. Simplify. Use a calculator.

  13. f Answer: Example 4 Find e and f.

  14. Summary & Homework • Summary: • The arithmetic mean of two numbers is their average (add and divide by two) • The geometric mean of two numbers is the square root of their product • You can use the geometric mean to find the altitude of a right triangle • Homework: • pg 346, 10, 11, 13-15, 21-24, 29-31

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