Two corners of a triangle have angles of # (5 pi )/ 12 # and # (3 pi ) / 8 #. If one side of the triangle has a length of # 15 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-06-25T10:30:20

The longest perimeter is #=61.6#

The third angle of the triangle is

#=pi-(5/12pi+3/8pi)#

#=pi-(10/24pi+9/24pi)#

#=pi-19/24pi=5/24pi#

The angles of the triangle in ascending order is

#5/12pi>9/24pi>5/24pi#

To get longest perimeter, we place the side of length #15# in font of the smallest angle, i.e. #5/24pi#

We apply the sine rule

#A/sin(5/12pi)=B/sin(3/8pi)=15/sin(5/24pi)=24.64#

#A=24.64*sin(5/12pi)=23.8#

#B=24.64*sin(3/8pi)=22.8#

The perimeter is

#P=15+23.8+22.8=61.6#