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#