Two corners of a triangle have angles of #(7 pi ) / 12 # and # pi / 6 #. If one side of the triangle has a length of #2 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-20T11:24:21

Longest possible perimeter P = 8.6921

Given #: /_ A = pi /6, /_B = (7pi)/12#

# /_C = (pi - pi /6 - (7pi)/12 ) = (pi)/4 #

To get the longest perimeter, we should consider the side corresponding to the angle that is the smallest.

#a / sin A = b / sin B = c / sin C#

#2 / sin (pi/6) = b / sin ((7pi)/12) = c / sin ((pi)/4)#

#:. b = (2 * sin ((7pi)/12)) / sin (pi/6) = 3.8637#

#c = (2 * sin (pi/4)) / sin (pi/6) = 2.8284#

Longest possible perimeter #P = 2 + 3.8637 + 2.8284 = 8.6921#