Two corners of a triangle have angles of #pi / 4 # and # pi / 3 #. If one side of the triangle has a length of #6 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-26T10:38:39

Longest possible perimeter of the triangle is 21.5447

Given #: /_ A = pi /4, /_B = (pi)/3#

# /_C = (pi - pi /4 - (pi)/3 ) = (5pi)/12 #

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#

#6 / sin (pi/4) = b / sin ((5pi)/12) = c / sin ((pi)/3)#

#:. b = (6 * sin ((5pi)/12)) / sin (pi/4) = 8.1962#

#c = (6 * sin (pi/3)) / sin (pi/4) = 7.3485#

Longest possible perimeter #P = 6 + 8.1962 + 7.3485 = 21.5447#