Two corners of a triangle have angles of #pi / 3 # and # pi / 2 #. If one side of the triangle has a length of #9 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-06-01T15:17:51

Longest possible perimeter #color(red)(P = 24.59# units

#hat A = pi/3, hat B = pi/2, hat C = pi - pi/3 - pi /2 = pi/6#

Side of length 9 should correspond to the least angle #pi/6# to get the longest perimeter.

Applying Law of Sines,

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

#a = (c sin B) / sin C = (9 * sin (pi/6)) / sin (pi/3) = 3 sqrt3#

#b = (9 sin(pi/2)) / sin (pi/3) = 6sqrt3#

Longest possible perimeter of the triangle is

#P = 9 + 6sqrt3 + 3sqrt3 = 24.59# units