Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 12 #. If one side of the triangle has a length of # 7 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-05-03T12:08:38

#color(brown)("Longest possible perimeter " P = 53.45 " sq units"#

#hat A = (5pi)/8, hat B = pi/12, hat C = pi - (5pi)/8 - pi/12 = (7pi)/24#

#color(blue)("As per Law of Sines,' color(crimson)(a / sin A = b / sin B = c / sin C#

To get the longest perimeter, side of length 7 should correspond to least angle #hat B = pi/12#

#:. a / sin ((5pi)/8) = 7 / sin (pi/12) = c / sin ((7pi)/24)#

#a = (7 * sin ((5pi)/8)) / sin (pi/12) ~~ 24.99#

#c = (7 sin ((7pi)/24)) / sin (pi/12) ~~ 21.46#

#color(brown)("Longest possible perimeter " P = 7 + 24.99 + 21.46 = 53.45#