Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 6 #. If one side of the triangle has a length of # 12 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-06-02T09:53:04

Longest possible perimeter of the triangle

#color(maroon)(P = a + b + c = 48.78#

#hat A = (5pi)/8, hat B = pi/6, hat C = pi - (5pi)/8 - pi/6 = (5pi)/24#

To get the longest perimeter, side 12 should correspond to the least angle #hat B = pi/6#

Applying the law of Sines,

#a = (b * sin A) / sin B = (12 sin ((5pi)/8))/sin (pi/6) = 22.17#

#c = (sin C * b) / sin B = (12 * sin ((5pi)/24))/sin (pi/6) = 14.61#

Longest possible perimeter of the triangle

#color(maroon)(P = a + b + c = 22.17+ 12 + 14.61 = 48.78#