Two corners of a triangle have angles of $( pi )/ 3$ and $( pi ) / 4$ . If one side of the triangle has a length of $5$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-02-19T06:11:55

Longest possible perimeter of the triangle is

$color(brown)(P = a + b + c ~~ 17.9538$

To find the longest possible perimeter of the triangle.

Given $hatA = pi/3, hatB = pi/4$ , one $side = 5$

$hatC = pi - pi/3 - pi/4 = (5pi)/12$

Angle $hatB$ will correspond to side 5 to get the longest perimeter.

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$a / sin A = b / sin B = c / sin C$ , applying sine law.

$a = (b sin A) / sin B = (5 * sin (pi/3)) / sin (pi/4) = 6.1237$

$c = (b sin C) / sin B = (5 * sin ((5pi)/12)) / sin (pi/4) = 6.8301$

Longest possible perimeter of the triangle is

$color(brown)(P = a + b + c = 6.1237 + 5 + 6.8301 ~~ 17.9538$

Two corners of a triangle have angles of ( pi )/ 3 ( pi )/ 3 and ( pi ) / 4 ( pi ) / 4 . If one side of the triangle has a length of 5 5 , what is the longest possible perimeter of the triangle?