Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 3 #. 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 · 2016-11-10T22:48:10

The longest possible perimeter is, #p = 58.8#

Let #angle C = (5pi)/8#
Let #angle B = pi/3#

Then #angle A = pi - angle B - angle C#

#angle A = pi - pi/3 - (5pi)/8#

#angle A = pi/24#

Associate the given side with the smallest angle, because that will lead to the longest perimeter:

Let side a = 4

Use the law of sines to compute the other two sides:

#b/sin(angleB) = a/sin(angleA) = c/sin(angleC) #

#b = asin(angleB)/sin(angleA) ~~ 26.5 #

#c = asin(angleC)/sin(angleA) ~~ 28.3#

#p = 4 + 26.5 + 28.3#

The longest possible perimeter is, #p = 58.8#