Two corners of a triangle have angles of $(7 pi )/ 12$ and $pi / 4$ . 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 · 2018-02-19T01:54:14

Longest possible perimeter of the triangle

$color(blue )(p = (a + b + c) = 39.1146)$

Given : $hatA = (7pi)/12, hatB = pi/4, side = 9$

Third angle is $hatC = pi - (7pi/12)/12 - pi/4 = pi/6$

To get the longest perimeter, least side should correspond to the smallest angle.

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By law of sines,

$a / sin A = b / sin B = c / sin C$

$:. a / sin (7pi)/12 = b / sin (pi/4) = 9 / sin (pi/6)$

Side $a = (9 * sin ((7pi)/12)) / sin (pi/6) = 17.3867$

Side $b = (9 * sin (pi/4)) / sin (pi/6) = 12.7279$

Longest possible perimeter of the triangle

$p = (a + b + c) = (17.3867 + 12.7279 + 9) = color(blue )(39.1146$

Two corners of a triangle have angles of (7 pi )/ 12 (7 pi )/ 12 and pi / 4 pi / 4 . If one side of the triangle has a length of 9 9 , what is the longest possible perimeter of the triangle?