Two corners of a triangle have angles of # (7 pi )/ 12 # and # pi / 4 #. If one side of the triangle has a length of 1, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-01-26T05:22:44

Longest possible perimeter of the triangle ABC is #color(green)(P = 4.3461)#

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Given #A = (7pi)/12, B = pi/4#

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

To get the largest perimeter, side 1 to correspond to least angle #pi/6#
We know,

#a / sin A = b / sin B = c / sin C#

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

#b = (1 * sin (pi/4)) / sin (pi/6) = 1.4142#

#c = (1 * sin ((7pi)/12)) / sin (pi/6) = 1.9319#

Perimeter of triangle, #P = (a + b + c)/2 #

#P = (1 + 1.4142 + 1.9319) = color(green)(4.3461)#