Answer: 16 + 16*sqrt(3) which is choice B
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Work Shown
arc HG = 120
angle HOG = 120
OH = OG = 8 = radius
angle OGH = (180-angle HOG)/2 = (180-120)/2 = 30
Use the law of sines
sin(angle HOG)/HG = sin(angle OGH)/OH
sin(120)/HG = sin(30)/8
(sqrt(3)/2)/HG = (1/2)/8
sqrt(3)/HG = 1/8
8*sqrt(3) = HG
HG = 8*sqrt(3)
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angle FOG = 180-(angle HOG) = 180-120 = 60
angle OGF = (180 - angle FOG)/2 = (180-60)/2 = 60
Triangle FOG is equilateral, therefore FO = GO = FG = 8
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Rectangle EFGH has length 8*sqrt(3) and width 8
P = perimeter of rectangle
P = 2*(length + width)
P = 2*(8*sqrt(3) + 8)
P = 16*sqrt(3) + 16
P = 16 + 16*sqrt(3)
The final answer is choice B
16 + 16*sqrt(3) = 43.7128 approximately