A triangle has two corners with angles of $\frac{3 π}{4}$ $(3 pi ) / 4$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $16$ $16$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{3 π}{4}$ $(3 pi ) / 4$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $16$ $16$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-02-15T07:22:31

Largest possible area of the triangle is $A_{t} \approx 174.85$ $A_t ~~ color (blue)(174.85$ sq units

Explanation:

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Given : $ˆ A = 3 \frac{π}{4}, B = \frac{π}{6},$ $hatA = 3pi/4, B = pi/6,$

Third angle $ˆ C = π - 3 \frac{π}{4} - \frac{π}{6} = \frac{π}{12}$ $hatC= pi - 3pi/4 -pi/6 = pi/12$

To get the longest area, length 16 should correspond to least angle $\frac{π}{12}$ $pi/12$

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $a / sin A = b / sin B = c / sin C$

$\frac{a}{sin (\frac{3 π}{4})} = \frac{b}{sin (\frac{π}{6})} = \frac{16}{sin (\frac{π}{12})}$ $a / sin((3pi)/4) = b / sin (pi/6) = 16 / sin (pi/12)$

$b = \frac{16 \cdot sin (\frac{π}{6})}{sin (\frac{π}{12})} \approx 30.91$ $b = (16 * sin (pi/6)) / sin (pi/12) ~~ 30.91$

Area of triangle $A_{t} = (\frac{1}{2}) b c sin ˆ A = (\frac{1}{2}) \cdot 30.91 \cdot 16 sin (\frac{3 π}{4})$ $A_t = (1/2) b c sin hatA = (1/2) * 30.91 * 16 sin ((3pi)/4)$

$\Rightarrow 174.85$ $=> 174.85$ sq units

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A triangle has two corners with angles of $\frac{3 π}{4}$ $(3 pi ) / 4$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $16$ $16$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of 3π4 (3 pi ) / 4 and π6 ( pi )/ 6 . If one side of the triangle has a length of 1616 , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of $\frac{3 π}{4}$ $(3 pi ) / 4$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $16$ $16$ , what is the largest possible area of the triangle?