Two corners of a triangle have angles of # (2 pi )/ 3 # and # ( pi ) / 6 #. If one side of the triangle has a length of # 16 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-01-07T03:15:25

Longest possible perimeter of the triangle is #color(purple)(P_t = 71.4256)#

Given angles #A = (2pi)/3, B = pi /6#

#C = pi - (2pi)/3 - pi/6 = pi/6#

It’s an isosceles triangle with sides b & c equal.

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To get the longest perimeter, smallest angle (B & C) should correspond to side 16

#a / sin ((2pi)/3) = 16 / sin (pi/6)#

#a = (16 * sin ((2pi)/3)) / sin (pi/6) = 27.7128#

Perimeter #P_t = a + b + c = 16 + 27.7128 + 27.7128 = color (purple)(71.4256)#

Longest possible perimeter of the triangle is #color(purple)(P_t = 71.4256)#