Two corners of a triangle have angles of # ( pi )/ 2 # and # ( pi ) / 6 #. If one side of the triangle has a length of # 14 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-05-12T04:41:05

#color(green)("Longest Possible Perimeter" = 14 + 24.25 + 28 = 66.25 " units"#

#hat A = pi/2, hat B = pi/6, hat C = pi - pi/2 - pi/6 = pi/3#

To get the longest perimeter, side 14 should correspond to the least angle #pi/6#

Applying Law of Sines,

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

#14 / sin (pi/6) = c / sin (pi/3)#

#c = (14 * sin (pi/3)) / sin (pi/6) = 24.25#

#a = (14 * sin(pi/2)) / sin (pi/6) = 28#

#color(green)(" Perimeter " P = a = b + c#

#color(green)("Longest Possible Perimeter" = 14 + 24.25 + 28 = 66.25 " units"#