Two corners of a triangle have angles of $(5 pi )/ 12$ and $( pi ) / 3$ . If one side of the triangle has a length of $1$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-06-02T11:20:07

Longest possible perimeter $color(orange)(P = 1 + 1.22 + 1.37 = 3.59$

$hat A = (5pi)/12, hat B = pi/3, hat C = pi/4$

Side 1 should correspond to $hat C = pi/4$ the least angle to get the longest perimeter.

As per Law of Sines, $a / sin A = b / sin B = c / sin C$

$:. a = (sin ((5pi)/12) * 1) / sin (pi/4) = 1.37$

$b = (sin (pi/3) * 1) / sin(pi/4) = 1.22$

Longest possible perimeter $color(orange)(P = 1 + 1.22 + 1.37 = 3.59$

Two corners of a triangle have angles of (5 pi )/ 12 (5 pi )/ 12 and ( pi ) / 3 ( pi ) / 3 . If one side of the triangle has a length of 1 1 , what is the longest possible perimeter of the triangle?