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

1 Answer
Oct 22, 2016

The perimeter is 61

Explanation:

Let #/_A = pi/12#
Let #/_B = (5pi)/8#
Then #/_C = pi - (5pi)/8 - pi/12 = (24pi)/24 - (15pi)/24 - (2pi)/24 = (7pi)/24#

Let side #a# be the side opposite #/_A# with length #8#. We do this because associating the given side with the smallest angle will yeild the largest perimeter.

Using the Law of Sines we can write the equation for sides #b# and #c#:

#b = asin(/_B)/sin(/_A)#
#c = asin(/_C)/sin(/_A)#

Let p = the perimeter

#p = a + a(sin(/_B) + sin(/_C))/sin(/_A)#

#p = 61#