Two corners of a triangle have angles of $\frac{2 π}{3}$ $(2 pi )/ 3$ and $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $13$ $13$ , 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 $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $13$ $13$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-10-16T02:22:53

Longest possible perimeter = 48.5167

Explanation:

$\frac{a}{sin a} = \frac{b}{sin b} = \frac{c}{sin c}$ $a/sin a = b/sin b = c/sin c$
The three angles are $\frac{2 π}{3}, \frac{π}{6}, \frac{π}{6}$ $(2pi)/3, pi/6, pi/6$

To get the longest possible perimeter, given side should correspond to the smallest angle $\frac{π}{6}$ $pi/6$

$\frac{13}{sin (\frac{π}{6})} = \frac{b}{sin (\frac{π}{6})} = \frac{c}{sin (\frac{2 π}{6})}$ $13/sin (pi/6) = b/sin (pi/6) = c/sin ((2pi)/6)$
$b = 13, c = ⎛ ⎜ ⎝ 13 \cdot ⎛ ⎜ ⎝ \frac{sin (\frac{2 π}{3})}{sin (\frac{π}{6})} ⎞ ⎟ ⎠$ $b=13, c = (13 * (sin ((2pi)/3)/sin (pi/6))$
$c = \frac{13 \cdot sin 120}{sin 60} = \frac{13 \cdot (\frac{\sqrt{3}}{2})}{\frac{1}{2}}$ $c = (13 * sin120)/sin 60 = (13 * (sqrt3/2))/(1/2)$
$sin (\frac{π}{6}) = \frac{1}{2}, sin (\frac{2 π}{3}) = sin (\frac{π}{3}) = \frac{\sqrt{3}}{2}$ $sin (pi/6) = 1/2, sin ((2pi)/3) = sin (pi/3) = sqrt3 / 2$
$c = 13 \cdot \sqrt{3} = 22.5167$ $c = 13*sqrt3=22.5167$

Perimeter $= 13 + 13 + 22.5167 = 48.5167$ $= 13+13+22.5167=48.5167$

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Two corners of a triangle have angles of $\frac{2 π}{3}$ $(2 pi )/ 3$ and $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $13$ $13$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Science

Math

Humanities

... and beyond

Two corners of a triangle have angles of 2π3 (2 pi )/ 3 and π6 ( pi ) / 6 . If one side of the triangle has a length of 13 13 , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of $\frac{2 π}{3}$ $(2 pi )/ 3$ and $\frac{π}{6}$ $( pi ) / 6$ . If one side of the triangle has a length of $13$ $13$ , what is the longest possible perimeter of the triangle?