Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 7 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-06-02T12:53:19

Perimeter #color(blue)( P = a + b + c = 36.83#

#hat A = (5pi)/8, hat B = pi/4, hat C = pi - (5pi)/8 - pi/4 = pi/8#

Least angle #hat C = pi/8# should correspond to the side 7 to get the longest perimeter.

Applying Law of Sines,

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

#a = (c sin A) / sin C = (7 * sin((5pi)/8)) / sin (pi/8) = 16.9#

#b = (7 * sin (pi/4)) / sin (pi/8) = 12.93#

Perimeter #color(blue)( P = a + b + c = 16.9 + 12.93 + 7 = 36.83#