Proposed Problem

Click the figure below to see the geometric illustration of Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square.

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Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square

Level: High School, SAT Prep, College geometry

## Saturday, July 4, 2009

### Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square

Labels:
arithmetic mean,
diameter,
geometric mean,
harmonic mean,
perpendicular

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1. a+b=OC+CB=AB=2.OD

ReplyDeleteSo OD=(a+b)/2= AM(a,b)

2. AEB is a right triangle. Relation in right triangle give CE^2=CA.CB= a.b

So CE= SQRT(a.b)=GM(a,b)

3. Relation in right triangle OCE give

EC^2=EF.EO => EF= EC^2/EO

EC^2=a.b , EO=DO=(a+b)/2

So EF=2a.b/(a+b)= HM(a,b)

4. In right triangle OCD we have CD^2=OD^2+OC^2

But OD=(a+b)/2 , OC=(a-b)/2

So CD^2=2(a^2+b^2) => CD=RMS(a,b)

5. In right triangles ODC and CEF we always have DC >OD and CE > EF

In circle O we always have OD > CE

So RMS (a,b) > AM (a,b) > GM (a,b) > HM (a,b)

6. AM(a,b).HM(a,b)=(a+b)/2 x (2.a.b)/(a+b)= ab= GM(a,b)