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 $12$ $12$ , what is the longest possible perimeter of the triangle?

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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 $12$ $12$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-06-03T09:47:02

Longest possible perimeter is $12 + 40.155 + 32.786 = 84.941$ $12+40.155+32.786=84.941$ .

Explanation:

As two angles are $\frac{2 π}{3}$ $(2pi)/3$ and $\frac{π}{4}$ $pi/4$ , third angle is $π - \frac{π}{8} - \frac{π}{6} = \frac{12 π - 8 π - 3 π}{24} \equiv \frac{π}{12}$ $pi-pi/8-pi/6=(12pi-8pi-3pi)/24-=pi/12$ .

For longest perimeter side of length $12$ $12$ , say $a$ $a$ , has to be opposite smallest angle $\frac{π}{12}$ $pi/12$ and then using sine formula other two sides will be

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

Hence $b = \frac{12 sin (\frac{2 π}{3})}{sin (\frac{π}{12})} = \frac{12 \times 0.866}{0.2588} = 40.155$ $b=(12sin((2pi)/3))/(sin(pi/12))=(12xx0.866)/0.2588=40.155$

and $c = \frac{12 \times sin (\frac{π}{4})}{sin (\frac{π}{12})} = \frac{12 \times 0.7071}{0.2588} = 32.786$ $c=(12xxsin(pi/4))/(sin(pi/12))=(12xx0.7071)/0.2588=32.786$

Hence longest possible perimeter is $12 + 40.155 + 32.786 = 84.941$ $12+40.155+32.786=84.941$ .

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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 $12$ $12$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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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 12 12 , what is the longest possible perimeter of the triangle?

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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 $12$ $12$ , what is the longest possible perimeter of the triangle?