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$ABCD$ is a square of side length 1. Points $E, F, G, H $ are such that $ AE : EB = 1:1$, $ BF: FC = 1 : 2 $, $ CG : GD = 1 : 3 $, $ DH : HA = 1 : 4 $.

What is the area of the quadrilateral $ EFGH $?

 

 Jan 29, 2021
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Note  that if AE : EB  = 1:1, then   EB  = 1/2

 

And if BF : BC  = 1 :2   , then  BF  = 1/3

 

So the area of right triangle   EBF  =(1/2) (1/2)(1/3)  =   1/12

 

Similarly  FC  =2/3

And if GC : GD  =1:3    then GC  = 1/4

So the area of right triangle FGC   = (1/2)(2/3)(1/4)  =  1/12

 

And GD = 3/4

And if HD : HA  =1 : 4   then   HD =  1/5

So the area of right trinagle  HDG  =(1/2) (3/4)(1/5)  = 3/40

 

And finally    HA =4/5    and AE =1/2

So the area of  right triangle HAE  =(1/2) (1/2) (4/5)  =  1/5

 

So....the blue area is    area of the square  - 1/12  -1/12  -  3/40  - 1/5  =

 

1  - 1/12 -1/12  - 3/40 - 1/5    =   67/120

 

 

cool cool cool

 Jan 29, 2021

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