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?
+129849 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
