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# help with geometry

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The figure shows congruent three squares aligned with the sides of an equilateral triangle. The inner sides of the squares form another equilateral triangle.  What is the ratio of the area of the smaller equilateral triangle to the larger equilateral triangle?

Dec 22, 2020

#1
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Let the side of the square  = S

Connect  the  bottom left vertex of the large equilateral triangle  with the vertex where the bottom square and leftmost square meet

This forms a  30-60-90   right traingle

We can find the  distance  from the bottom left vertex of  the  large equilateral square to the bottom left vertex of the bottom square  =  S* sqrt (3)

So....the  side of  the  equilateral triangle is   ( 2sqrt(3)S  + S ) =  S ( 2sqrt (3) + 1) =  S [ sqrt (12) + 1 ]

And the side of  the  smaller  equilateral triangle    = S

The triangles are  similar figures  and  their scale factor  =  [ S ]  / [ S *( 2sqrt (3) + 1) ]  =  1 / [ sqrt (12) + 1 ]

The ratio of the area of the  smaller equilateral triangle to the larger = [ scale factor] ^2   =

[ 1 / [sqrt (12) + 1 ] ]^2    =

1 / [ 12 + 2sqrt (12)  + 1 ]  =

1/ [ 13  + 2sqrt (12) ]   =

1/  [ 13  + sqrt (48) ]   =

13 -sqrt (48)

___________  =

169   - 48

13 -sqrt (48)

_________  =

121

13  - 2sqrt (3)

___________   ≈  .05  ≈  1/20

121

Dec 22, 2020