Two corners of a triangle have angles of $\frac{3 π}{8}$ $(3 pi ) / 8$ and $\frac{π}{4}$ $pi / 4$ . If one side of the triangle has a length of $4$ $4$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-11T06:00:40

Longest possible perimeter of the triangle is 14.4526

Given are the two angles $\frac{π}{4}$ $(pi)/4$ and $\frac{3 π}{8}$ $(3pi)/8$ and the length 1

The remaining angle:

$= π - ((\frac{π}{4}) + \frac{3 π}{8}) = \frac{3 π}{8}$ $= pi - (((pi)/4) + (3pi)/8) = (3pi)/8$

I am assuming that length AB (4) is opposite the smallest angle

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

$\frac{4}{sin (\frac{π}{4})} = \frac{b}{sin (\frac{3 π}{8})} = \frac{c}{\frac{3 π}{8}}$ $4 / sin ((pi)/4) = b / sin ((3pi) /8) = c / ((3pi) / 8)$

$b = \frac{4 \cdot sin (\frac{3 π}{8})}{sin (\frac{π}{4})} = 5.2263$ $b = (4*sin((3pi)/8)) / sin ((pi) /4) = 5.2263$

$c = \frac{4 \cdot sin (\frac{3 π}{8})}{sin (\frac{π}{4})} = 5.2263$ $c = (4*sin ((3pi)/8)) / sin ((pi)/4) = 5.2263$

Longest possible perimeter of the triangle is = $(a + b + c) = (4 + 5.2263 + 5.2263) = 14.4526$ $(a+b+c) = (4+5.2263+5.2263) = 14.4526$

Two corners of a triangle have angles of 3π8(3 pi ) / 8 and π4 pi / 4 . If one side of the triangle has a length of 44 , what is the longest possible perimeter of the triangle?