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In the figure, A and B are the midpoints of two sides of a regular hexagon.  What fraction of the hexagon's area is gray?  Express your answer as a common fraction.

 

 Jul 18, 2022
 #1
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Let the midpoint of the base  of  the triangle  =  (0,0)

Let the bottom left vertex of the triangle =  (0, -1)

Let  the top left vertex o f the hexagon be   ( -1/2 , sqrt (3)  / 2)

A =  [   (-1 + -1/2)  /2   ,  (sqrt (3)/2 + 0)/ 2 ] =  [ -3/4  , sqrt (3) / 4) ] 

 

So   the  base of the triangle =  2    and the height = y coord of A  =   sqrt (3) / 4

Its area =   (1/2) (2) ( sqrt (3) / 4)  =    sqrt (3)  /4

 

Area of  the hexagon =   (6) ( 1)^2  * sin (60°)  =   3 sqrt (3)  / 2 =     6sqrt (3) / 4 

 

Fraction that is  gray =       sqrt (3) / 4                 1

                                      ____________  =        ____

                                        6 sqrt (3) / 4                 6

 

cool cool cool

 Jul 18, 2022

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