Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $12$ $12$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-04-01T07:36:42

$color(green)("Longest possible perimeter of the ") color(indigo)(Delta = 91.62 " units"$

$hat A = (5pi)/8, hat B = pi/12, hat C = pi - (5pi)/8 - pi/12 = (7pi)/24$

To find the longest possible perimeter of the triangle, we length 12 should correspond to side b as $hat B$ has the least angle measure.

Applying the Law of Sines,

$a / sin A = b / sin B = c / sin C$

$a = (12 * sin ((5pi)/8))/ sin (pi/12) = 42.84 " units"$

$c = (12 * sin((7pi)/24)) / sin (pi/12) = 36.78 " units"$

$"Longest possible perimeter of the " Delta = (a + b + c)$

$=> 42.84 + 36.78 + 12 = 91.62 " units"$

Two corners of a triangle have angles of (5 pi )/ 8 5π8 (5 pi )/ 8 and ( pi ) / 12 π12 ( pi ) / 12 . If one side of the triangle has a length of 12 12 12 , what is the longest possible perimeter of the triangle?