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# Geometry Question

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The field shown has been planted uniformly with wheat. At harvest, the wheat at any point in the field is brought to the nearest point on the field’s perimeter. What is the fraction of the crop that is brought to the longest side?

- Daisy

Aug 23, 2018
edited by dierdurst  Aug 23, 2018

#1
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I'm not sure....but here's my best effort

Let ABCD  be the  trapezoidal field

Its  area   =

(1/2)(height) (DC + AB) =

(1/2)(50√3) (100 + 200) =

7500√3   m^2

And  the  crop will be brought to the longer side whenever it is contained within  the trapezoid  AEFB

Reasoning :

DB bisects  angle CBA....angle FIB  = FKB  and FB  = FB

So....by AAS, triangle FIB  = triangle FKB

So....any crop harvested within triangle FIB will be closer to IB than BK

For the same reason any crop harvested within triangle EJA  will be closer to JA  than side AL

And any crop harvested within EFIJ  will be closer to the longest side than to side DC

And the area of AEFB  =

(1/2)(EJ) (AB + EF)  =

(1/2)(25√3) (200 + 50) =

3125√3  m^2

So  the fraction of the crop brought to the longest side  is

3125√3  / 7500√3  =   5/12

Aug 24, 2018
#2
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Thank you!

- Daisy

dierdurst  Aug 24, 2018