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
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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)