Two corners of a triangle have angles of #(7 pi ) / 12 # and # pi / 6 #. If one side of the triangle has a length of #6 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-07-08T10:59:28

THe longest perimeter is #=26.1u#

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Let

#hatA=7/12pi#

#hatB=1/6pi#

So,

#hatC=pi-(7/12pi+1/6pi)=1/4pi#

The smallest angle of the triangle is #=1/6pi#

In order to get the longest perimeter, the side of length #6#

is #b=6#

We apply the sine rule to the triangle #DeltaABC#

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

#a/sin (7/12pi) = c/ sin(1/4pi)=6/sin(1/6pi)=12#

#a=12*sin (7/12pi)=11.6#

#c=12*sin(1/4pi)=8.5#

The perimeter of triangle #DeltaABC# is

#P=a+b+c=11.6+6+8.5=26.1#