A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{7 π}{8}$ $(7 pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , 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 $\frac{7 π}{8}$ $(7 pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-01-07T02:54:40

Largest possible area of the triangle is $A_{t} = 64.1194$ $color(blue)(A_t = 64.1194)$

Explanation:

Given $∠ A = \frac{7 π}{8}, ∠ B = \frac{π}{12}$ $/_A = (7pi)/8, /_B = pi /12$

$∠ C = π - \frac{7 π}{8} - \frac{π}{12} = \frac{π}{24}$ $/_C = pi - (7pi)/8 - pi/12 = pi / 24$

Smallest angle is $∠ C = \frac{π}{24}$ $/_C = pi/24$

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To get the largest area of the triangle possible, smallest angle should correspond to the given length 13.

$i . e c = 13$ $i.e c = 13$

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

Hence,
$\frac{a}{sin (\frac{7 π}{8})} = \frac{b}{sin (\frac{π}{12})} = \frac{13}{sin (\frac{π}{24})}$ $a / sin ((7pi)/8) = b / sin (pi/12) = 13 / sin (pi / 24)$

$a = \frac{13 \cdot sin (\frac{7 π}{8})}{sin (\frac{π}{24})} = 38.1141$ $a = (13 * sin ((7pi)/8)) / sin (pi/24) = 38.1141$

$b = \frac{13 \cdot sin (\frac{π}{12})}{sin (\frac{π}{24})} = 25.7776$ $b = (13 * sin (pi/12)) / sin (pi / 24) = 25.7776$

Area of triangle $A_{t}$ $A_t$ =1/2 . Base . Height

Base $a = 38.1141$ $a = 38.1141$

Height $h = c sin (∠ B) = 13 \cdot sin (\frac{π}{12}) = 3.3646$ $h = c sin (/_B) = 13 * sin (pi / 12) = 3.3646$

$A_{t} = (\frac{1}{2}) \cdot 38.1141 \cdot 3.3646 = 64.1194$ $A_t = (1/2) * 38.1141 * 3.3646 = color(blue)(64.1194)$

Largest possible area of the triangle is $A_{t} = 64.1194$ $color(blue)(A_t = 64.1194)$

Verification :

$* A_{t} * = (\frac{1}{2}) \cdot a \cdot b sin C = (\frac{1}{2}) \cdot 38.1141 \cdot 25.7776 \cdot sin (\frac{π}{24}) = * 64.1194 *$ $**A_t** = (1/2) * a * b sin C = (1/2) * 38.1141 * 25.7776 * sin (pi/24) =**64.1194**$

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A triangle has two corners with angles of $\frac{π}{12}$ $pi / 12$ and $\frac{7 π}{8}$ $(7 pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , 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 $\frac{7 π}{8}$ $(7 pi )/ 8$ . If one side of the triangle has a length of $13$ $13$ , what is the largest possible area of the triangle?