A triangle has two corners with angles of # pi / 4 # and # pi / 6 #. If one side of the triangle has a length of #16 #, what is the largest possible area of the triangle?

1 Answer
Feb 13, 2018

largest possible area of the triangle #color(purple)(A_t ~~ 699.405#

Explanation:

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Third angle #C = pi - pi/4 - pi/6 = (7pi)/12#

To get the largest possible area, length 16 should correspond to least angle #pi/6#

Other sides are

#a / sin(pi/4) = b / sin (7pi)/12 = c / sin (pi/6) = 16 / sin (pi/6) = 32# as #sin(pi/6) = sin 30 = 1/2#

#a = 32 sin (pi/4) = 32 sqrt2 = 45.2548#

Area of triangle #A_t = (1/2) a c sin B = cancel(1/2) cancel(32)^color(red)(16) sqrt2 * 32 * sin ((7pi)/12)#

#A_t = 512 sqrt2 sin ((7pi)/12) = color(purple)(699.405# sq units