Geometric and Arithmetic Means

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Presentation transcript:

Geometric and Arithmetic Means 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 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

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 √ (6•16) = √96 ≈ 9.8 (4+8)/2 = 6 √ (4•8) = √32 ≈ 5.66 (5+10)/2 = 7.5 √ (5•10) = √50 ≈ 7.07 (2+14)/2 = 8 √ (2•14) = √28 ≈ 5.29

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.

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

Application of Geometric Mean b altitude hypotenuse (length = a + b) From similar triangles a x --- = --- x 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

Example of Geometric Mean 6 14 x altitude 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. x = √6 • 14 = √84 ≈ 9.17

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

Example 2b Answer: about 8.5

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

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

Example 4 Find e and f. f Answer:

Summary & Homework Summary: Homework: 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