A triangle has two corners with angles of # pi / 12 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #1 #, what is the largest possible area of the triangle?

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Answer 1 · 2017-12-05T09:22:42

Area of largest possible triangle #= color (red)(2.0056)#

Three angles are #pi/12, (5pi)/8, (pi - (pi/12) + ((5pi)/8) =( 7pi)/24#

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

To get the largest possible are, smallest angle should correspond to the side of length 1.

#1 / sin (pi/12) = b / sin ((7pi)/24) = c / sin ((5pi)/8)#

#b = (sin ((7pi)/24)) / (sin (pi/12)#
#b = 3.0653#

#c = (sin ((5pi)/8)) / (sin (pi/12))#
#c = 3.5696#

Semi perimeter #s = (a + b + c) / 2 = (1+3.0653+3.5696)/2 = 3.8175#

#s-a = 3.8175-1 = 2.8175#
#s-b = 3.8175-3.0653 = 0.7522#
#s-c = 3.8175-3.5696 = 0.2479#

Area of #Delta = sqrt(s (s-a) (s-b) (s-c))#

#Area of Delta = sqrt(3.8175 * 2.8175 * 0.7522 * 0.2479) = color (red)(2.0056)#