A triangle has sides A, B, and C. Sides A and B have lengths of 4 and 5, respectively. The angle between A and C is #(7pi)/24# and the angle between B and C is # (3pi)/8#. What is the area of the triangle?

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Answer 1 · 2015-12-15T15:28:28

#5 sqrt 3#

We use ABC for points; and a,b,c for opposite sides.

angle between a and c = #hat B = 7/24 pi#

angle between b and c = #hat A = 3/8 pi#

#hat C = pi - hat B - hat A = pi (1 - 7/24 - 3/8) = 1/3 pi#

#c^2 = a^2 + b^2 - 2 ab cos hat C#

#c^2 = 16 + 25 - 20#

#c = sqrt 21#

#S_Delta = sqrt {p (p-a)(p-b)(p-c)}#

#p = (9 + sqrt 21)/2#

#p - 4 = (1 + sqrt 21)/2#

#p - 5 = (-1 + sqrt 21)/2#

#p - sqrt 21 = (9 - sqrt 21)/2#

#S_Delta^2 = (9 + sqrt 21)/2 * (1 + sqrt 21)/2 * (-1 + sqrt 21)/2 * (9 - sqrt 21)/2#

#16 * S_Delta^2 = (81 - 21) * (21 - 1)#

#S_Delta =sqrt{ (60 * 20) / 16} = 10/4 sqrt 12 = 5/2 * 2 sqrt 3#