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An equilateral triangle is constructed on each side of a square with side length $2,$ as shown below.  The four outer vertices are then joined to form a large square.  Find the area of the shaded region.

 Dec 14, 2023
 #1
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Nothing shown, but I assume you want the area of the large square ???

 

Let G = (0,0)

Let HG = 1

Using the Pythagorean Theorem, the height of one of the equilateral triangles  = sqrt (EA^2 - HA^2)  = sqrt ( 2^2 -1^2)  = sqrt (3)

So  E = ( 0, HG + HE)  = (0, 1 + sqrt (3))

And by symmetry, F =  (1 + sqrt 3, 0)

The distance between  these points = EF = the side of the large square  =  sqrt [ (1 + sqrt 3)^2 + (1 + sqrt 3)^2 ] =

sqrt [ 2 * ( 1 + sqrt 3)^2 ]  

 

So....the area of the larger square = EF^2  =   2 * (1 + sqrt 3)^2  =  2 * (4 +2sqrt 3)  = 

 

8 + 4sqrt 3  = 

 

 4 ( 2 + sqrt 3)

 

 

cool cool cool

 Dec 14, 2023

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