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If the blue triangles continue to be formed indefinitely, what fraction of the big square will be shaded blue?

 

 Feb 1, 2021
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Call the side  of the square, S

 

Area of largest blue  triangle  =   (S)^2 / 2   =  S^2 /2

 

Area of  next largest blue triangle =   (S/2)^2 / 2 =  S^2  / 8

 

This  infinite series  continues   as  S^2 /32  ,etc. (each successive  square has 1/4  the area of the previous one = the  common  ratio )

 

So  the sum of the areas  is  S^2 (1/2  +  1/8   + 1/32  + ....... )

 

The sum of the series inside the parentheses is given  as     first term  /( 1 - common ratio )

 

(1/2)  / ( 1  -1/4)   =    (1/2) / (3/4)  =  (1/2) (4/3)  =  4 / 6  = 2/3

 

So

 

The fraction of the square  covered =    2/3

 

cool cool cool

 Feb 1, 2021

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