Two corners of a triangle have angles of # (2 pi )/ 3 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 19 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-01-20T15:24:24

Longest possible perimeter

#color(green)(P = 19 + 51.909 + 63.5752 = 134.4842)#

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Three angles are #(2pi) /3, pi/4, pi/12# as the three angles add up to #pi^c#

To get the longest perimeter, side 19 should correspond to the smallest angle #pi/12#

#19 / sin (pi/12) = b / sin (pi/4) = c / sin ((2pi)/3)#

#b = (19 * sin (pi/4)) / sin (pi/12) = 51.909#

#c = (19 * sin ((2pi)/3)) / sin (pi/12) = 63.5752#

Longest possible perimeter

#color(green)(P = 19 + 51.909 + 63.5752 = 134.4842)#