A triangle has two corners with angles of # ( pi ) / 4 # and # ( 7 pi )/ 12 #. If one side of the triangle has a length of #7 #, what is the largest possible area of the triangle?

1 Answer
Oct 25, 2016

Largest possible area of the triangle is #33.467#

Explanation:

As two angles of a triangle are #pi/4# and #(7pi)/12#, the third angle is

#pi-pi/4-(7pi)/12=(12pi-3pi-7pi)/12=(2pi)/12=pi/6#

All triangles with these angles are similar. As length of one side is #7#, its area will be maximum, if this side is opposite smallest angle i.e. #pi/6#.

Area of a triangle given one side #a=7# and two angles #/_A=pi/6#, #/_B=pi/4# and #/_C=(7pi)/12# is

#(a^2sinBsinC)/(2sinA)#

= #(7^2sin((7pi)/12)sin(pi/4))/(2sin(pi/6))#

= #(49xx0.9659xx0.7071)/(2xx0.5)#

= #(49xx0.0.683)/1#

= #33.467#