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 $5$ $5$ , 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 $5$ $5$ , what is the largest possible area of the triangle?

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Answer 1 · 2017-10-18T03:48:13

Area of triangle $A = \sqrt{s (s - a) (s - b) (s - c)}$ $A = sqrt(s (s-a) (s-b) (s - c))$

$* A = 145.1551 *$ $**A = 145.1551**$

Explanation:

Three angles are $\frac{π}{12}, \frac{π}{8}, \frac{19 π}{24}$ $pi/12, pi/8, (19pi)/24$
$\frac{a}{sin a} = \frac{b}{sin b} = \frac{c}{sin c}$ $a/sin a = b / sin b = c / sin c$
$\frac{5}{sin (\frac{π}{12})} = \frac{b}{sin (\frac{π}{8})} = \frac{c}{sin (\frac{19 π}{24})}$ $5/sin (pi/12) = b/sin (pi/8) = c /sin ((19pi)/24)$

$b = \frac{5 \cdot sin (\frac{π}{8})}{sin (\frac{π}{12})} = 7.3929$ $b = (5* sin (pi/8))/sin (pi/12) = 7.3929$

$c = ⎛ ⎜ ⎝ 5 \cdot \frac{sin (\frac{19 π}{24})}{sin (\frac{π}{12})} = 11.7604$ $c = (5* sin ((19pi)/24)/sin (pi/12)=11.7604$

$s = \frac{a + b + c}{2} = \frac{5 + 7.3928 + 11.7604}{2} = 18.273$ $s = (a + b + c) /2 = (5+7.3928+11.7604)/2= 18.273$
$s - a = 13.273$ $s-a = 13.273$
$s - b = 10.8801$ $s-b = 10.8801$
$s - c = 6.5126$ $s-c = 6.5126$

Area of triangle $A = \sqrt{s (s - a) (s - b) (s - c)}$ $A = sqrt(s (s-a) (s-b) (s - c))$

$A = \sqrt{18.273 \cdot 13.273 \cdot 10.8801 \cdot 6.5126} = 145.1551$ $A = sqrt(18.273 * 13.273 * 10.8801 * 6.5126) = 145.1551$

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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 $5$ $5$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{8}$ $pi / 8$ . If one side of the triangle has a length of $5$ $5$ , what is the largest possible area of the triangle?