Two corners of a triangle have angles of # (5 pi )/ 12 # and # pi / 6 #. If one side of the triangle has a length of 3, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-02-05T11:51:24

Perimeter of the longest possible triangle is #14.6# unit.

Angle between Sides # A and B# is #

#/_c= (5pi)/12=(5*180)/12=75^0#

Angle between Sides # B and C# is # /_a= pi/6=180/6=30^0 :.#

Angle between Sides # C and A# is

# /_b= 180-(75+30)=75^0#. For largest perimeter of

triangle #3# should be smallest side , which is opposite

to the smallest angle #/_a=30^0:.A=3#. 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#

#:. A/sina=B/sinb or 3/sin30 = B/sin 75: B = (3*sin75)/sin30# or

#B~~5.80 ; B/sinb=C/sinc or 5.80/sin75=C/sin75#

#:. C~~ 5.8 :. A=3.0 , B~~ 5.8 , C~~ 5.8 # . Perimeter of the

triangle is #P_t=A+B+C ~~ 3.0+5.8+5.8=14.6 # unit.

Perimeter of the longest possible triangle is #14.6# unit [Ans]