A triangle has two corners with angles of # (2 pi ) / 3 # and # ( pi )/ 6 #. If one side of the triangle has a length of #15 #, what is the largest possible area of the triangle?

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Answer 1 · 2017-11-13T11:29:25

Largest possible area of #Delta = 97.43#

The three angles are #(2pi)/3, (pi)/6 , pi - (2pi)/3 - pi/6 = pi/6#
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It’s an isosceles triangle.

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

# 15 / sin (pi/6) = b / sin ((2pi)/3) = c / sin (pi/6)#

#:. c = 15#

#b = (15 * sin (2pi/3)) / sin (pi/6)#

# b = (15 * (sqrt3/2))/ (1/2) = 15 sqrt3#

Height #h = 15 * sin( pi /6) = 15/2#

Area of #Delta = (1/2) b h#

Area of #Delta = (1/2) * 15 sqrt3 * (15/2)#

#= (225 sqrt3) / 4 = 97.43#