Two corners of a triangle have angles of #pi / 8 # and # pi / 6 #. If one side of the triangle has a length of #2 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-01-24T09:59:14

Longest possible perimeter is #color(brown )((2 + 2.6131 + 4.1463) = 8.7594)#

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Given : #alpha = pi/8, eta = pi/6, gamma = pi - (pi/8 + pi/6) = ((17pi)/24)#

To get the longest perimeter, length ‘2’ should correspond to side ‘a’ which is opposite to the smallest angle #alpha#

Three sides are in the ratio,

#a / sin alpha = b / sin beta = c / sin gamma#

#b = (2 * sin beta) / sin alpha = (2*sin(pi/6)) / sin (pi/8)#

#b = (2 * (1/2)) / sin (pi/8) ~~ 2.6131#

Similarly,
#c = (2 * sin ((17pi)/24)) / sin (pi/8) ~~ 4.1463#

Longest possible perimeter is #color(brown )((2 + 2.6131 + 4.1463) = 8.7594)#