Two corners of a triangle have angles of # (5 pi )/ 12 # and # ( pi ) / 3 #. 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 · 2017-12-08T09:34:20

Longest possible perimeter = 32.3169

Sum of the angles of a triangle #=pi#

Two angles are #(5pi)/12, pi/3#
Hence #3^(rd) #angle is #pi - ((5pi)/12 + pi/3) = pi/4#

We know# a/sin a = b/sin b = c/sin c#

To get the longest perimeter, length 2 must be opposite to angle #pi/4#

#:. 9/ sin(pi/4) = b/ sin((5pi)/12) = c / sin ((pi)/3)#

#b = (9 sin((5pi)/12))/sin (pi/4) = 12.2942#

#c =( 9* sin((pi)/3))/ sin (pi/4) = 11.0227#

Hence perimeter #= a + b + c = 9 + 12.2942 + 11.0227 = 32.3169#