Two corners of a triangle have angles of $\frac{2 π}{3}$ $(2 pi )/ 3$ and $\frac{π}{4}$ $( pi ) / 4$ . If one side of the triangle has a length of $19$ $19$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-01-20T15:24:24

Longest possible perimeter

$P = 19 + 51.909 + 63.5752 = 134.4842$ $color(green)(P = 19 + 51.909 + 63.5752 = 134.4842)$

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Three angles are $\frac{2 π}{3}, \frac{π}{4}, \frac{π}{12}$ $(2pi) /3, pi/4, pi/12$ as the three angles add up to $π^{c}$ $pi^c$

To get the longest perimeter, side 19 should correspond to the smallest angle $\frac{π}{12}$ $pi/12$

$\frac{19}{sin (\frac{π}{12})} = \frac{b}{sin (\frac{π}{4})} = \frac{c}{sin (\frac{2 π}{3})}$ $19 / sin (pi/12) = b / sin (pi/4) = c / sin ((2pi)/3)$

$b = \frac{19 \cdot sin (\frac{π}{4})}{sin (\frac{π}{12})} = 51.909$ $b = (19 * sin (pi/4)) / sin (pi/12) = 51.909$

$c = \frac{19 \cdot sin (\frac{2 π}{3})}{sin (\frac{π}{12})} = 63.5752$ $c = (19 * sin ((2pi)/3)) / sin (pi/12) = 63.5752$

Longest possible perimeter

$P = 19 + 51.909 + 63.5752 = 134.4842$ $color(green)(P = 19 + 51.909 + 63.5752 = 134.4842)$

Two corners of a triangle have angles of 2π3 (2 pi )/ 3 and π4 ( pi ) / 4 . If one side of the triangle has a length of 19 19 , what is the longest possible perimeter of the triangle?