Two corners of a triangle have angles of #(3 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 · 2017-07-07T12:15:46

The longest perimeter is #=75.6u#

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Let

#hatA=3/8pi#

#hatB=1/12pi#

So,

#hatC=pi-(3/8pi+1/12pi)=13/24pi#

The smallest angle of the triangle is #=1/12pi#

In order to get the longest perimeter, the side of length #9#

is #b=9#

We apply the sine rule to the triangle #DeltaABC#

#a/sin hatA=c/sin hatC=b/sin hatB#

#a/sin (3/8pi) = c/ sin(13/24pi)=9/sin(1/12pi)=34.8#

#a=34.8*sin (3/8pi)=32.1#

#c=34.8*sin(13/24pi)=34.5#

The perimeter of triangle #DeltaABC# is

#P=a+b+c=32.1+9+34.5=75.6#