Two corners of a triangle have angles of # (7 pi )/ 12 # and # pi / 4 #. 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 · 2018-02-19T01:54:14

Longest possible perimeter of the triangle

#color(blue )(p = (a + b + c) = 39.1146)#

Given : #hatA = (7pi)/12, hatB = pi/4, side = 9#

Third angle is #hatC = pi - (7pi/12)/12 - pi/4 = pi/6#

To get the longest perimeter, least side should correspond to the smallest angle.

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By law of sines,

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

#:. a / sin (7pi)/12 = b / sin (pi/4) = 9 / sin (pi/6)#

Side #a = (9 * sin ((7pi)/12)) / sin (pi/6) = 17.3867#

Side #b = (9 * sin (pi/4)) / sin (pi/6) = 12.7279#

Longest possible perimeter of the triangle

#p = (a + b + c) = (17.3867 + 12.7279 + 9) = color(blue )(39.1146#