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# Geometry

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Can I get a hint?

A square has its vertices on the edges of a regular hexagon.  Two of the edges of the square are parallel to two edges of the hexagon, as shown in the diagram.  The sides of the hexagon have length 1.

What is the length of the sides of the square?

Dec 22, 2023

#1
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Let O  be the center of the square

Let M  be the top right vertex of the  square

Let N be the rightmost vertex of the hexagon  and P  be the top right vertex of the  hexagon

Triangle  PON is equilateral  so   PN =  ON  =  1

OM  =  (1/2) diagonal of square =   S /sqrt 2       where S is the side of the square

Angle  OMN  = 75°     Angle MON =  45°    Angle MNO = 60°

Using the Law of Sines

sin (OMN) / 1  =  sin MNO / (S/sqrt 2)

sin (75) /1 =  sin (60)/ (S/ sqrt 2)

S / sqrt (2) =  sin 60 / sin 75

S =    sqrt (2) sin (60) / sin (75)

S = sqrt (2) (sqrt (3) / 2) /  ( sin (45 + 30)

S = sqrt (6) / (2 (sin 45cos 30  + cos 45 sin 30) )

S =  sqrt (6) / ( 2 ( sqrt (2)/2 * sqrt (3)/2  + sqrt(2/2 )(1/2) )

S =  sqrt (6) / (2 (sqrt (6) /4  + sqrt (2)  / 4))

S = sqrt (6) / [ (sqrt (6) + sqrt (2)) / 2

S = 2 sqrt (6) / (sqrt 6 + sqrt 2)

S = 2 sqrt (6) ( sqrt 6 - sqrt 2) / ( 6-2)

S =  (12 - 2sqrt (12)) /4

S =  3 - 2sqrt (4) sqrt (3) / 4

S =  3 - 2*2 * sqrt (3) / 4

S = 3 - 4sqrt (3) / 4

S =  3 - sqrt (3)  ≈  1.27

Here's a pic  (AM =  1/2 the side of the square ≈  .63 )

Dec 22, 2023