Two corners of a triangle have angles of # (2 pi )/ 3 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 8 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-26T15:39:11

Longest possible perimeter of triangle is #56.63# unit.

Angle between Sides # A and B# is # /_c= (2pi)/3=120^0#

Angle between Sides # B and C# is # /_a= pi/4=45^0 :.#

Angle between Sides # C and A# is

# /_b= 180-(120+45)=15^0#

For longest perimeter of triangle #8# should be smallest side,

the opposite to the smallest angle , #:. B=8#

The sine rule states if #A, B and C# are the lengths of the sides

and opposite angles are #a, b and c# in a triangle, then:

#A/sina = B/sinb=C/sinc ; B=8 :. B/sinb=C/sinc# or

#8/sin15=C/sin120 or C= 8* (sin120/sin15) ~~ 26.77 (2dp) #

Similarly #A/sina=B/sinb # or

#A/sin45=8/sin15 or A= 8* (sin45/sin15) ~~ 21.86 (2dp) #

Longest possible perimeter of triangle is #P_(max)=A+B+C# or

#P_(max)=26.77+8+ 21.86 ~~ 56.63# unit [Ans]