A triangle has two corners with angles of # pi / 12 # and # pi / 4 #. If one side of the triangle has a length of #14 #, what is the largest possible area of the triangle?

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Answer 1 · 2018-02-28T12:27:18

Longest possible perimeter #Perimeter# #P = color(purple)(84.9374#

Given #hat A = pi / 12, hat B = pi / 4, hatC = pi - pi/12 - pi/4 = (2pi)/3#

To get the longest perimeter, side 14 should correspond to the least #hatA#.

Applying law of sines,
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#12 / sin (pi/12) = b / sin (pi/4) = c / sin ((2pi)/3)#

#b = (12 * sin (pi/4)) / sin (pi/12) = 32.7846#

#c = (12 * sin ((2pi)/3)) / sin (pi/12) = 40.1528#

Perimeter #P = 12 + 32.7846 + 40.1528 = color(purple)(84.9374#