Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 12 #. If one side of the triangle has a length of # 1 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-11-23T20:02:25

The longest possible perimeter is #P ~~ 10.5#

Let #angle A = pi/12#
Let #angle B = (5pi)/8#
Then #angle C = pi - (5pi)/8 - pi/12#
#angle C = (7pi)/24#

The longest perimeter occurs, when the given side is opposite the smallest angle:

Let side #a = "the side opposite angle A" = 1#

The perimeter is: #P = a + b + c#

Use the Law of Sines

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

to substitute into the perimeter equation:

#P = a(1 + sin(B) + sin(C))/sin(A)#

#P = 1(1 + sin((5pi)/8) + sin((7pi)/24))/sin(pi/12)#

#P ~~ 10.5#