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

1 Answer
Apr 1, 2018

#color(green)("Longest possible perimeter of the ") color(indigo)(Delta = 91.62 " units"#

Explanation:

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

To find the longest possible perimeter of the triangle, we length 12 should correspond to side b as #hat B# has the least angle measure.

Applying the Law of Sines,

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

#a = (12 * sin ((5pi)/8))/ sin (pi/12) = 42.84 " units"#

#c = (12 * sin((7pi)/24)) / sin (pi/12) = 36.78 " units"#

#"Longest possible perimeter of the " Delta = (a + b + c)#

#=> 42.84 + 36.78 + 12 = 91.62 " units"#