A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $6$ $6$ , what is the largest possible area of the triangle?

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $6$ $6$ , what is the largest possible area of the triangle?

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Answer 1 · 2017-10-19T22:02:57

Largest possible area of triangle = 262.5667

Explanation:

Three angles are $\frac{π}{12}, \frac{π}{8}, \frac{17 π}{24}$ $pi/12, pi/8, (17pi)/24$

To get larges area, length 6 must correspond to least angle $\frac{π}{12}$ $pi/12$

$\frac{6}{sin (\frac{π}{12})} = \frac{b}{sin (\frac{π}{8})} = \frac{c}{sin (\frac{19 π}{24})}$ $6/ sin (pi/12) = b / sin (pi/8) = c / sin ((19pi)/24)$

$b = \frac{6 \cdot \frac{sin (π)}{8}}{sin (\frac{π}{12})} = 8.8715$ $b = (6 * sin (pi)/8) / sin (pi/12) = 8.8715$

$c = \frac{6 \cdot sin (\frac{19 π}{24})}{sin (\frac{π}{12})} = 14.1124$ $c = (6*sin ((19pi)/24)) / sin (pi/ 12) = 14.1124$

Semiperimeter $s = \frac{a + b + c}{2} = \frac{6 + 8.8715 + 14.1124}{2} = 14.492$ $s = ( a + b + c)/2 = (6+8.8715+14.1124)/2 = 14.492$

$s - a = 14.492 - 6 = 8.492$ $s-a = 14.492 - 6 = 8.492$
$s - b = 14.492 - 8.8715 = 5.6205$ $s-b = 14.492 - 8.8715 = 5.6205$
$s - c = 14.492 - 14.1124 = 0.3796$ $s-c = 14.492 - 14.1124 = 0.3796$

Area of $Delta = sqrt(s (s-a) (s-b) (s-c))$

Area of $Delta = sqrt(14.492 * 8.492 * 5.6205 * 0.3796) = **262.5667**$

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $6$ $6$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of pi / 12 π12 pi / 12 and pi / 8 π8 pi / 8 . If one side of the triangle has a length of 6 66 , what is the largest possible area of the triangle?

1 Answer

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $6$ $6$ , what is the largest possible area of the triangle?