A triangle has sides A, B, and C. If the angle between sides A and B is $\frac{π}{12}$ $(pi)/12$ , the angle between sides B and C is $\frac{2 π}{3}$ $(2pi)/3$ , and the length of B is 20, what is the area of the triangle?

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Answer 1 · 2015-12-15T15:59:09

$100 \sqrt{6} \cdot sin \frac{π}{12}$ $100 sqrt 6 * sin frac{pi }{12}$

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

angle between a and b = $ˆ C = \frac{1}{12} π$ $hat C = 1/12 pi$

angle between b and c = $ˆ A = \frac{2}{3} π$ $hat A = 2/3 pi$

$ˆ B = π - ˆ C - ˆ A = π (1 - \frac{1}{12} - \frac{2}{3}) = \frac{1}{4} π$ $hat B = pi - hat C - hat A = pi (1 - 1/12 - 2/3) = 1/4 pi$

Let $H \in A C$ $H in AC$ , such that $B H$ $BH$ is perpendicular to $A C$ $AC$ .

$| B H | = h, | A H | = m$ $|BH| = h, |AH| = m$ and we want $S_Delta = 1/2 * 20 * h$

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$tan hat A = h / m Rightarrow m = h / tan hat A$

$tan hat C = h / (20 - m) Rightarrow 20 - m = h / tan hat C$

$20 - h / tan hat A = h / tan hat C$

$20 = h ( 1 / tan hat A + 1 / tan hat C)$

$20 = h ( cos hat A / sin hat A + cos hat C / sin hat C)$

$h = 20 * frac{sin A sin C}{sin A cos C + sin C cos A} = 20/ sin frac{3 pi}{4} * sin frac{2 pi }{3} sin frac{pi}{12}$

$S_Delta = 10h = 200 * 2/sqrt 2 * sqrt 3 / 2 * sin frac{pi }{12}$

A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/12π12(pi)/12, the angle between sides B and C is (2pi)/32π3(2pi)/3, and the length of B is 20, what is the area of the triangle?