Two corners of a triangle have angles of $\frac{5 π}{12}$ $(5 pi )/ 12$ and $\frac{π}{3}$ $( pi ) / 3$ . If one side of the triangle has a length of $6$ $6$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-08T12:32:03

Largest possible area of the triangle is 21.2942

Given are the two angles $\frac{5 π}{12}$ $(5pi)/12$ and $\frac{π}{3}$ $pi/3$ and the length 6

The remaining angle:

$= π - ((\frac{5 π}{12}) + \frac{π}{3}) = \frac{π}{4}$ $= pi - (((5pi)/12) + pi/3) = pi/4$

I am assuming that length AB (6) is opposite the smallest angle.

Using the ASA

Area $= \frac{c^{2} \cdot sin (A) \cdot sin (B)}{2 \cdot sin (C)}$ $=(c^2*sin(A)*sin(B))/(2*sin(C)$

Area $= \frac{6^{2} \cdot sin (\frac{π}{3}) \cdot sin (\frac{5 π}{12})}{2 \cdot sin (\frac{π}{4})}$ $=( 6^2*sin(pi/3)*sin((5pi)/12))/(2*sin(pi/4))$

Area $= 21.2942$ $=21.2942$

Two corners of a triangle have angles of 5π12 (5 pi )/ 12 and π3 ( pi ) / 3 . If one side of the triangle has a length of 6 6 , what is the longest possible perimeter of the triangle?