Two corners of a triangle have angles of #(3 pi ) / 4 # 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 · 2018-05-20T06:39:41

Longest possible perimeter of the triangle is

#color(blue)(= 39.64# units

#hat A = (3pi)/4, hat B = pi/12, hat C = pi - (3pi)/4 - pi/12 = pi/6#

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

Applying Law of Sines,

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

#a/ sin ((3pi)/4) = 7 / sin (pi/12) = c / sin (pi/6)#

#a = (7 * sin((3pi)/4))/sin(pi/12) =19.12#

#c = (7*sin(pi/6))/sin(pi/12) = 13.52#

Perimeter #= a + b+ c = 19.12+ 7 + 13.52 = 39.64#