Geometric and Arithmetic Means

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
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 v(410) v40 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
(616)/2 11
(48)/2 6
(510)/2 7.5
(214)/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)
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
8
Example of Geometric Mean
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.
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
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Example 3
Cross products
Divide each side by 5.
12
Example 3 cont
Cross products
Take the square root.
Simplify.
Use a calculator.
13
Example 4
Find e and f.
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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