Two corners of a triangle have angles of #pi / 8 # and # pi / 4 #. If one side of the triangle has a length of #4 #, what is the longest possible perimeter of the triangle?

1 Answer
Apr 10, 2016

Longest possible perimeter: #~~21.05#

Explanation:

If two of the angles are #pi/8# and #pi/4#
the third angle of the triangle must be #pi - (pi/8+pi/4) = (5pi)/8#

For the longest perimeter, the shortest side must be opposite the shortest angle.
So #4# must be opposite the angle #pi/8#

By the Law of Sines
#color(white)("XXX")("side opposite "rho)/(sin(rho)) = ("side opposite " theta)/(sin(theta))# for two angles #rho# and #theta# in the same triangle.

Therefore
#color(white)("XXX")#side opposite #pi/4=(4*sin(pi/4))/(sin(pi/8)) ~~7.39#
and
#color(white)("XXX")#side opposite #(5pi)/8 = (4*sin((5pi)/8))/(sin(pi/8))~~9.66#

For a total (maximum) perimeter of
#color(white)("XXX")4+7.39+9.66= 21.05#