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In regular octagon ABCDEFGH,  M and  N are midpoints of  line BC and  line FG respectively. Compute [ABMO]/[EDCMO]. ([ABCD] denotes the area of polygon ABCD.)

 

 

Thanks in advance!!

MIRB14  Jan 30, 2018
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 #1
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O will be the center of the octagon

 

Draw OB, OC  and OD

 

Note that   area OAB  =  area  ODC =  area ODE

 

And area OBM  =  1/2 area OAB

 

And area OCM  = area OBM  =  1/2 area OAB

 

So

 

area  EDCMO  =   area ODE + area ODC  + area OCM

 

Substituting, we have that

 

area EDCMO  =  area OAB + area OAB + 1/2area OAB   =  2.5 areaOAB

 

 

And area ABMO  =  area OAB + area OBM

 

Substituting, we have that

 

area ABMO  =  area OAB + 1/2 area OAB   =   1.5 area OAB

 

So

 

[ ABMO] / [ EDCMO]  =

 

[ 1.5 area OAB ] / [ 2.5 area OAB ]  =

 

[1.5] / [ 2.5]  =

 

3 / 5 

 

 

cool cool cool

CPhill  Jan 30, 2018

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