[b]Definition: [/b][br][br]The [color=#9900ff][b]GEOMETRIC MEAN[/b][/color] of two numbers [i][b][color=#0000ff]a[/color][/b] [/i]and [i][color=#980000][b]b[/b][/color][/i] is defined to be the square root of their product. [br]That is, the [b][color=#9900ff]GEOMETRIC MEAN[/color][/b] of [i][b][color=#0000ff]a[/color][/b] [/i]and [i][color=#980000][b]b[/b][/color][/i] = [math]\sqrt{ab}[/math]. [br][br]In the applet below, the [b][color=#9900ff]purple segment that will soon appear[/color][/b] is the [b][color=#9900ff]GEOMETRIC MEAN[/color][/b] of [i][b][color=#0000ff]a[/color][/b] [/i]and [i][b][color=#980000]b[/color][/b][/i][b]. [br][br][/b][b]Interact with the applet for a few minutes. [/b]Be sure to change the lengths of [i][b][color=#0000ff]a[/color][/b] [/i]and [i]b [/i]as you do. [br]Then, answer the questions that follow.
What is the [color=#9900ff][b]geometric mean[/b][/color] of 5 and 20?
What is the [b][color=#9900ff]geometric mean[/color][/b] of 4 and 5?
12 is the [b][color=#9900ff]geometric mean[/color][/b] of 36 and what other number?
Prove that the purple segment drawn in the semicircle above has a length equal to the geometric mean of [i][b][color=#0000ff]a[/color][/b] [/i]and [b][color=#980000][i]b[/i]. [/color][/b]