Two corners of a triangle have angles of # (2 pi )/ 3 # and # ( pi ) / 6 #. If one side of the triangle has a length of # 4 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-08T09:39:03

Longest possible perimeter = 14.928

Sum of the angles of a triangle #=pi#

Two angles are #(2pi)/3, pi/6#
Hence #3^(rd) #angle is #pi - ((2pi)/3 + pi/6) = pi/6#

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

To get the longest perimeter, length 2 must be opposite to angle #pi/24#

#:. 4/ sin(pi/6) = b/ sin((pi)/6) = c / sin ((2pi)/3)#

#b = (4 sin((pi)/6))/sin (pi/6) = 4#

#c =( 4* sin((2pi)/3))/ sin (pi/6) = 6.9282#

Hence perimeter #= a + b + c = 4 + 4 + 6.9282 = 14.9282#