A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{4}$ $pi / 4$ . If one side of the triangle has a length of $14$ $14$ , 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 $\frac{π}{4}$ $pi / 4$ . If one side of the triangle has a length of $14$ $14$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-02-28T12:27:18

Longest possible perimeter $P e r i m e t e r$ $Perimeter$ $P = 84.9374$ $P = color(purple)(84.9374$

Explanation:

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

To get the longest perimeter, side 14 should correspond to the least $ˆ A$ $hatA$ .

Applying law of sines,
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$\frac{12}{sin (\frac{π}{12})} = \frac{b}{sin (\frac{π}{4})} = \frac{c}{sin (\frac{2 π}{3})}$ $12 / sin (pi/12) = b / sin (pi/4) = c / sin ((2pi)/3)$

$b = \frac{12 \cdot sin (\frac{π}{4})}{sin (\frac{π}{12})} = 32.7846$ $b = (12 * sin (pi/4)) / sin (pi/12) = 32.7846$

$c = \frac{12 \cdot sin (\frac{2 π}{3})}{sin (\frac{π}{12})} = 40.1528$ $c = (12 * sin ((2pi)/3)) / sin (pi/12) = 40.1528$

Perimeter $P = 12 + 32.7846 + 40.1528 = 84.9374$ $P = 12 + 32.7846 + 40.1528 = color(purple)(84.9374$

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{π}{4}$ $pi / 4$ . If one side of the triangle has a length of $14$ $14$ , 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 / 4 π4 pi / 4 . If one side of the triangle has a length of 14 1414 , 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 $\frac{π}{4}$ $pi / 4$ . If one side of the triangle has a length of $14$ $14$ , what is the largest possible area of the triangle?