Two corners of a triangle have angles of #(3 pi ) / 8 # and # pi / 8 #. If one side of the triangle has a length of #8 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-05T12:11:22

Longest possible perimeter of the #Delta# is #color(red)(54.2174)#

Three angles are #pi/8, (3pi)/8, pi/2#

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

To get the largest possible are, smallest angle should correspond to the side of length 8

#8/ sin (pi/8) = b / sin ((pi)/2) = c / sin ((3pi)/8)#

#b = (8*sin (pi/2)) / sin (pi / 8) = (8*1) / (0.3827 ) = 20.9041#

#c = (8* sin ((3pi)/8)) / sin (pi/8) = (8 * (0.9239))/(0.3827) = 19.3133#

Perimeter = a + b + c = 14 + 20.9041 + 19.3133 = color (red)(54.2174)#