Two corners of a triangle have angles of # (5 pi )/ 12 # and # ( pi ) / 3 #. If one side of the triangle has a length of # 15 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-02-19T05:21:52

Longest possible perimeter

#p = a + b + c ~~ color (green)(53.86#

To longest possible perimeter of the triangle.

Given : #hatA = (5pi)/12, hatB = pi/3#, one #side = 15#

Third angle #hatC = pi - (5pi)/12 - pi/3 = pi/4#

To get the longest perimeter, side 15 should correspond to the smallest angle #hatC = pi/4#

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Using sine law, #a/ sin A = b / sin B = c / sin C#

#a / sin (5pi)/12 = b / sin(pi/3) = 15 / sin (pi/4)#

#a = (15 * sin ((5pi)/12)) / sin (pi/4) ~~ 20.49#

#b = (15 * sin (pi)/3) / sin (pi/4) ~~ 18.37#

Longest possible perimeter

#p = a + b + c = 20.49 + 18.37 + 15 = color (green)(53.86#