Difference between revisions of "Geometric mean"

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The geometric mean arises in [[geometry]] in the following situation: if <math>AB</math> is a [[chord]] of [[circle]] <math>O</math> with [[midpoint]] <math>M</math> and <math>M</math> divides the [[diameter]] passing through it into pieces of length <math>a</math> and <math>b</math> then the length of [[line segment]] <math>AM</math> is the geometric mean of <math>a</math> and <math>b</math>.
 
The geometric mean arises in [[geometry]] in the following situation: if <math>AB</math> is a [[chord]] of [[circle]] <math>O</math> with [[midpoint]] <math>M</math> and <math>M</math> divides the [[diameter]] passing through it into pieces of length <math>a</math> and <math>b</math> then the length of [[line segment]] <math>AM</math> is the geometric mean of <math>a</math> and <math>b</math>.
{{image}}
+
 
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<asy>
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size(150);
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pointfontsize=8;
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pathfontsize=8;
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pair A=(3,4),B=(3,-4),M=(3,0);
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D((-5,0)--(5,0)); D(M--B);
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MC("\sqrt{ab}",D(A--M,orange+linewidth(1)),W);
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MC("a",D((-5,-0.3)--(3,-0.3),black,Arrows),S);
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MC("b",D((3,-0.3)--(5,-0.3),black,Arrows),S);
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D(CR(D((0,0)),5));
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D("A",A,N); D("B",B);D("M",M,NE);
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</asy>
  
 
The geometric mean also arises in the following common [[word problem]]: if a driver travels half the distance of a trip at a speed of <math>a</math> miles per hour and the other half at a speed of <math>b</math> miles per hour, the average speed over the whole trip is the geometric mean of <math>a</math> and <math>b</math>.  (If the driver spent half the ''time'' of the trip at each speed, we would instead get the arithmetic mean.)
 
The geometric mean also arises in the following common [[word problem]]: if a driver travels half the distance of a trip at a speed of <math>a</math> miles per hour and the other half at a speed of <math>b</math> miles per hour, the average speed over the whole trip is the geometric mean of <math>a</math> and <math>b</math>.  (If the driver spent half the ''time'' of the trip at each speed, we would instead get the arithmetic mean.)

Revision as of 18:49, 2 April 2008

The geometric mean of a collection of $n$ positive real numbers is the $n$th root of the product of the numbers. Note that if $n$ is even, we take the positive $n$th root. It is analogous to the arithmetic mean (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers $b$ and $c$ is the number $a$ such that $a + a = b + c$, while the geometric mean of the numbers $b$ and $c$ is the number $g$ such that $g\cdot g = b\cdot c$.

Examples

The geometric mean of the numbers 6, 4, 1 and 2 is $\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}$.

The geometric mean features prominently in the Arithmetic Mean-Geometric Mean Inequality.

The geometric mean arises in geometry in the following situation: if $AB$ is a chord of circle $O$ with midpoint $M$ and $M$ divides the diameter passing through it into pieces of length $a$ and $b$ then the length of line segment $AM$ is the geometric mean of $a$ and $b$.

[asy] size(150); pointfontsize=8; pathfontsize=8; pair A=(3,4),B=(3,-4),M=(3,0); D((-5,0)--(5,0)); D(M--B);  MC("\sqrt{ab}",D(A--M,orange+linewidth(1)),W); MC("a",D((-5,-0.3)--(3,-0.3),black,Arrows),S); MC("b",D((3,-0.3)--(5,-0.3),black,Arrows),S); D(CR(D((0,0)),5)); D("A",A,N); D("B",B);D("M",M,NE); [/asy]

The geometric mean also arises in the following common word problem: if a driver travels half the distance of a trip at a speed of $a$ miles per hour and the other half at a speed of $b$ miles per hour, the average speed over the whole trip is the geometric mean of $a$ and $b$. (If the driver spent half the time of the trip at each speed, we would instead get the arithmetic mean.)

Practice Problems

Introductory Problems

See Also

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