Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-01-24T09:59:14

Longest possible perimeter is $(2 + 2.6131 + 4.1463) = 8.7594$ $color(brown )((2 + 2.6131 + 4.1463) = 8.7594)$

Explanation:

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Given : $α = \frac{π}{8}, η = \frac{π}{6}, γ = π - (\frac{π}{8} + \frac{π}{6}) = (\frac{17 π}{24})$ $alpha = pi/8, eta = pi/6, gamma = pi - (pi/8 + pi/6) = ((17pi)/24)$

To get the longest perimeter, length ‘2’ should correspond to side ‘a’ which is opposite to the smallest angle $α$ $alpha$

Three sides are in the ratio,

$\frac{a}{sin α} = \frac{b}{sin β} = \frac{c}{sin γ}$ $a / sin alpha = b / sin beta = c / sin gamma$

$b = \frac{2 \cdot sin β}{sin α} = \frac{2 \cdot sin (\frac{π}{6})}{sin (\frac{π}{8})}$ $b = (2 * sin beta) / sin alpha = (2*sin(pi/6)) / sin (pi/8)$

$b = \frac{2 \cdot (\frac{1}{2})}{sin (\frac{π}{8})} \approx 2.6131$ $b = (2 * (1/2)) / sin (pi/8) ~~ 2.6131$

Similarly,
$c = \frac{2 \cdot sin (\frac{17 π}{24})}{sin (\frac{π}{8})} \approx 4.1463$ $c = (2 * sin ((17pi)/24)) / sin (pi/8) ~~ 4.1463$

Longest possible perimeter is $(2 + 2.6131 + 4.1463) = 8.7594$ $color(brown )((2 + 2.6131 + 4.1463) = 8.7594)$

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Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Two corners of a triangle have angles of π8pi / 8 and π6 pi / 6 . If one side of the triangle has a length of 22 , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{π}{8}$ $pi / 8$ and $\frac{π}{6}$ $pi / 6$ . If one side of the triangle has a length of $2$ $2$ , what is the longest possible perimeter of the triangle?