Two corners of a triangle have angles of #pi / 8 # and # pi / 12 #. If one side of the triangle has a length of #9 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-13T12:56:01

Longest possible perimeter is #43.48# unit.

Angle between Sides # A and B# is # /_c= pi/8=180/8=22.5^0#

Angle between Sides # B and C# is # /_a= pi/12=180/12=15^0 :.#

Angle between Sides # C and A# is

# /_b= 180-(22.5+15)=142.5^0# For largest perimeter of

triangle #9# should be smallest side , which is opposite to

smallest angle #:.A=9# unit.

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=9 :. A/sina=B/sinb# or

#9/sin 15=B/sin142.5 or B= 9* (sin142.5/sin15) ~~ 21.17 (2dp) #

Similarly #A/sina=C/sinc # or

#9/sin15=C/sin22.55 or C= 9* (sin22.5/sin15) ~~ 13.31 (2dp) #

Perimeter #P=A+B+C =9+21.17+13.31 ~~43.48# unit.

Longest possible perimeter is #43.48# unit. [Ans]