Two corners of a triangle have angles of # (7 pi )/ 12 # and # pi / 8 #. If one side of the triangle has a length of # 8 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-04-19T07:37:30

#color(brown)("Longest possible perimeter " = 8 + 20.19 + 16.59 = 44.78#

#hat A = (7pi)/12, hat B = pi/8, hat C = pi - (7pi)/12 - pi/8 = (7pi)/24#

To get the longest perimeter, side 8 should correspond to the least angle #pi/8#

Applying the Law of Sines,

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

#a / sin ((7pi)/12) = 8 / sin (pi/8) = c / sin ((7pi) / 24)#

#a = (8 * sin ((7pi)/12)) / sin (pi/8) ~~ 20.19#

#c = (8 * sin ((7pi)/24)) / sin (pi/8) ~~ 16.59#

#color(brown)("Longest possible perimeter " = 8 + 20.19 + 16.59 = 44.78#