Two corners of a triangle have angles of # (7 pi )/ 12 # 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 · 2018-02-19T05:49:11

Longest possible perimeter of the triangle is

#color(blue)(P + a + b + c ~~ 34.7685#

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#hatA = (7pi)/12, hatB = pi/4, side = 8#

To find the longest possible perimeter of the triangle.

Third angle #hatC = pi - (7pi)/12 - pi/4 = pi/6#

To get the longest perimeter, smallest angle #hatC = pi/6# should correspond to side length 8#

Using sine law, #a / sin A = b / sin B = c / sin C#

#a = (c * sin A) / sin C = (8 * sin((7pi)/12)) / sin (pi/6) = 15.4548#

#b = (c * sin B) / sin C = (8 * sin(pi/4) )/ sin (pi/6) = 11.3137#

Longest possible perimeter of the triangle is

#color(blue)(P + a + b + c = 15.4548 + 11.3137 + 8 = 34.7685#