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# Unique Problem

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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 side length of the large square.

Originally I got the answer of 2√2, but that was incorrect (just a word of warning).

May 12, 2023

#1
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If you draw the altitude of that top triangle, you create a 1, sqrt(3), 2 right triangle.

That altitude is sqrt(3), then coming on down, the line through the square is 2,

then you've got another sqrt(3) at the bottom.  So the diagonal is 2 + (2)*sqrt(3).

Let's simplify that to 2 * (1 + sqrt(3))

The diagonal of a square is the square root of 2 times a side.

So, a side is the diagonal divided by the square root of 2.

The side of the large square:

2 * (1 + sqrt(3))

————————

sqrt(2)

You could divide that sqrt(2) on the bottom into the 2 on the top, to

get (sqrt(2) * (1 + sqrt(3)) and multiply that out, do what you want.

.

May 12, 2023
#2
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In th top triangle, the angle of the large square is 90 degrees.

The angle of the equilateral triangle ==60 and splits the right angle into 3 angles:

[90 - 60] / 2 ==15 degrees - the size of the 2 smaller angles

So, on any of the 4 sides of the larger square, you have 4 highly obtuse isosceles triangles:

With sides: 2, 2 and the large obtuse angle between them==150 degrees

Use the law of cosines to get the side of the square==3.864

May 12, 2023