Two corners of a triangle have angles of $(5 pi )/ 8$ and $( pi ) / 3$ . If one side of the triangle has a length of $9$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2017-12-05T10:43:03

Longest possible perimeter $= color (purple)(132.4169)$

Sum of the angles of a triangle $=pi$

Two angles are $(5pi)/8, pi/3$
Hence $3^(rd)$ angle is $pi - ((5pi)/8 + pi/3) = pi/24$

We know $a/sin a = b/sin b = c/sin c$

To get the longest perimeter, length 9 must be opposite to angle $pi/24$

$:. 9/ sin(pi/24) = b/ sin((5pi)/8) = c / sin (pi/3)$

#b = (9 sin((5pi)/8))/sin (pi/24) = 63.7030

$c =( 9* sin(pi/3))/ sin (pi/24) = 59.7139$

Hence perimeter $= a + b + c = 9 + 63.7030 + 59.7139 = 132.4169$

Two corners of a triangle have angles of (5 pi )/ 8 (5 pi )/ 8 and ( pi ) / 3 ( pi ) / 3 . If one side of the triangle has a length of 9 9 , what is the longest possible perimeter of the triangle?