Two corners of a triangle have angles of $\frac{5 π}{12}$ $(5 pi )/ 12$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $9$ $9$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{5 π}{12}$ $(5 pi )/ 12$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $9$ $9$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-11-23T21:16:41

$P = 9 (3 + \sqrt{3} + \sqrt{6} + \sqrt{2}) \approx 77.36$ $P=9(3+sqrt3+sqrt6+sqrt2)approx77.36$ .

Explanation:

In $△ A B C$ $triangleABC$ , let $A = \frac{5 π}{12}, B = \frac{π}{12}$ $A=(5pi)/12,B=pi/12$ . Then

$C = π - A - B$ $C=pi-A-B$
$C = \frac{12 π}{12} - \frac{5 π}{12} - \frac{π}{12}$ $C=(12pi)/12-(5pi)/12-pi/12$
$C = \frac{6 π}{12} = \frac{π}{2}$ $C=(6pi)/12=pi/2$ .

In all triangles, the shortest side is always opposite the shortest angle. Maximizing the perimeter means putting the largest value we know (9) in the smallest position possible (opposite $∠ B$ $angleB$ ). Meaning for the perimeter of $△ A B C$ $triangleABC$ to be maximized, $b = 9$ $b=9$ .

Using the law of sines, we have

$\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}$ $sinA/a=sinB/b=sinC/c$

Solving for $a$ $a$ , we get:

$a=(bsinA)/sinB=(9sin((5pi)/12))/sin(pi/12)=(9(sqrt6+sqrt2)//4)/((sqrt6-sqrt2)//4)=...=9(2+sqrt3)$

Similarly, solving for $c$ yields

$c=(bsinC)/sinB=(9sin(pi/2))/(sin(pi/12))=(9(1))/((sqrt6-sqrt2)//4)=...=9(sqrt6+sqrt2)$

The perimeter $P$ of $triangleABC$ is the sum of all three sides:

$P=color(orange)a+color(blue)b+color(green)c$
$P=color(orange)(9(2+sqrt3))+color(blue)9+color(green)(9(sqrt6+sqrt2))$
$P=9(2+sqrt3+1+sqrt6+sqrt2)$
$P=9(3+sqrt3+sqrt6+sqrt2)approx77.36$

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Two corners of a triangle have angles of $\frac{5 π}{12}$ $(5 pi )/ 12$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $9$ $9$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of (5 pi )/ 12 5π12 (5 pi )/ 12 and ( pi ) / 12 π12 ( pi ) / 12 . If one side of the triangle has a length of 9 9 9 , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of $\frac{5 π}{12}$ $(5 pi )/ 12$ and $\frac{π}{12}$ $( pi ) / 12$ . If one side of the triangle has a length of $9$ $9$ , what is the longest possible perimeter of the triangle?