A triangle has sides A, B, and C. If the angle between sides A and B is #(pi)/4#, the angle between sides B and C is #(7pi)/12#, and the length of B is 5, what is the area of the triangle?

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Answer 1 · 2017-10-29T18:00:06

Area#=17.075units^2#

#cancelpi^color(magenta)1/cancel4^color(magenta)1xxcancel180^color(magenta)45/cancelpi^color(magenta)1=45^@=anglec#

#(7cancelpi^color(magenta)1)/cancel12^color(magenta)1xxcancel180^color(magenta)15/cancelpi^color(magenta)1=105^@=anglea#

#180^@-(105^@+45^@)=30^@=angleb#

#A/(sinanglea)=B/(sinangleb)#

#A=(sin105^@xx5)/sin30^@#

#A=4.829629131/0.5#

#A=9.659258262#

#color(magenta)(A=9.659#units to the nearest 3 decimal places

#C/(sinanglec)=B/(sinangleb)#

#C=(5*sin45^@)/sin30^@#

#C=3.535533906/0.5#

#C=7.071067812#

#color(magenta)(C=7.071units# to the nearest 3 decimal places

Area#=1/2BCsinanglea#

#0.5*5*7.071067812*sin105^@#

Area#=17.07531755units^2#

Area#color(magenta)(=17.075units^2# to the nearest 3 decimal places

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check:-

Hero's formula:-

#S=(a+b+c)/2#

#S=(9.659+5+7.071)/2#

Area#=sqrt(s(s-a)(s-b)(s-c))#

#S=10.865#units

Area#sqrt(10.865(10.865-9.659)(10.865-5)(10.865-7.071))#

#Area=sqrt(10.865(1.206)(5.865)(3.794))#

#Areacolor(magenta)(=17.075units^2# to the nearest 3 decimal places