A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 25, what is the area of the triangle?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 25, what is the area of the triangle?

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Answer 1 · 2018-05-03T11:42:49

$Area of Triangle A_{t} = 57.2 sq units$ $color(maroon)("Area of Triangle " A_t = 57.2 " sq units"$

Explanation:

$Given ˆ A = \frac{π}{12}, ˆ C = \frac{π}{6}, ˆ B = π - \frac{π}{12} - \frac{π}{6} = \frac{3 π}{4}, b = 25$ $color(blue)("Given " hat A = pi/12, hat C = pi/6, hat B = pi - pi/12 - pi/6 = (3pi)/4, b = 25$

$As per Law of sines, \frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $"As per " color(red)("Law of Sines", color(green)(a / sin A = b / sin B = c / sin C$

$\frac{a}{sin (\frac{π}{12})} = \frac{25}{sin (\frac{3 π}{4})} = \frac{c}{sin (\frac{π}{6})}$ $a / sin (pi/12) = 25 / sin ((3pi)/4) = c / sin (pi/6)$

$c = \frac{25 \cdot sin (\frac{π}{6})}{sin (\frac{3 π}{4})} = 17.68$ $c = (25 * sin (pi/6)) / sin ((3pi)/4) = 17.68$

$"Area of " Delta = A_t = (1/2) *b * sin A$

$A_t = (1/2) * 25 * 17.68 * sin (pi/12) = 57.2 " sq units"$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 25, what is the area of the triangle?

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A triangle has sides A, B, and C. The angle between sides A and B is pi/6π6pi/6 and the angle between sides B and C is pi/12π12pi/12. If side B has a length of 25, what is the area of the triangle?

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Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 25, what is the area of the triangle?