Geometric Mean

Geometric Mean

Geometric Mean is "an average which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum)." ~Wikipedia

The Setup

The geometric mean is the nth root of the product of n numbers. We can use a proportion to do this. If x is the mean of numbers a and b, then... $\frac{a}{x}=\frac{x}{b}$  .

 

Solve

To solve, we cross multiply and take the square root. 
 $ab=x^2$  
 $\sqrt{ab}=x$  

Multiple Choice

Find the geometric mean of 8 and 12.

• 9.7

• 10

• 96

• 9.8

Multiple Choice

Find the geometric mean of 15 and 200.

• 54.8

• 107.5

• 100

• 76.1

Multiple Choice

Find the arithmetic mean and the geometric mean of 10 and 17.4.

• 13.2 and 13.7

• 26.5 and 28.1

• 11.8 and 16.4

• 13.7 and 13.2

Now there's a triangle...

We can also find the geometric mean from a right triangle with an altitude drawn from the vertex.

THREE Right Triangles!

If you draw an altitude from the vertex of a right triangle to the opposite side, then the result is three similar right triangles.

Shaving Man

Put stars by the man's "eyes" and "nose"; these are your means.

In this figure, b and a are the "eyes" and are the means. Also, h is the "nose" and is a mean.

Shaving Man's Left Side Shave

Circle b, x, and c. This is a "left side shave". You are starting from his "ear" (b) and shaving all the way down to his chin (c).

These are the 3 values that you will use in your proportion.

Since b is the mean, b will appear twice in the proportion.

Shaving Man's Left Side Shave

 $\frac{x}{b}=\frac{b}{c}$  

Shaving Man's Right Side Shave

Circle a, y, and c. This is a "right side shave". You are starting from his "ear" (a) and shaving all the way down to his chin (c).

These are the 3 values that you will use in your proportion.

Since a is the mean, a will appear twice in the proportion.

Shaving Man's Right Side Shave

 $\frac{y}{a}=\frac{a}{c}$  

Shaving Man's Mustache Shave

Circle x, h, and y. This is a "mustache shave". You are starting from his "cheek" (x), shaving under his nose (h), and shaving back down his other cheek (y).

These are the 3 values that you will use in your proportion.

Since h is the mean, h will appear twice in the proportion.

Shaving Man's Mustache Shave

 $\frac{x}{h}=\frac{h}{y}$  

Let's Shave!

Multiple Choice

What type of shave would help us solve for x?

• Right Side Shave

• Left Side Shave

• Mustache Shave

• What??

Multiple Choice

What proportion would help us solve for x?

•  $\frac{4}{13}=\frac{9}{x}$  

•  $\frac{4}{x}=\frac{x}{9}$  
  

•  $\frac{9}{4}=\frac{4}{x}$  
  

• Not enough information

Multiple Choice

Solve for x.

• x = 8

• x = 6

• x = 7

• x = 5

Multiple Choice

What type of shave would help us to solve for x?

• Mustache Shave

• Right Side Shave

• Left Side Shave

• Not enough information

Multiple Choice

What proportion would help us to solve for x?

• $\frac{x}{6}=\frac{6}{13.5}$

• $\frac{6}{x}=\frac{x}{6}$

• $\frac{6}{x}=\frac{x}{13.5}$

• $\frac{x}{13.5}=\frac{13.5}{6}$

Multiple Choice

Solve for x.

• x = 6

• x = 7

• x = 8

• x = 9