A triangle has sides A, B, and C. The angle between sides A and B is #(3pi)/4#. If side C has a length of #9 # and the angle between sides B and C is #pi/12#, what are the lengths of sides A and B?

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Answer 1 · 2016-07-06T11:33:18

#A=9/2(sqrt(3)-1)#; #B=9/2sqrt(2)#

You can use the equation:

#A/sin hat(BC)=C/sin hat(AB)#

that, in this case, is:

#A/sin(pi/12)=9/sin(3pi/4)#

#A=9((sqrt(6)-sqrt(2))/4)/(sqrt(2)/2)#

#A=9((sqrt(6)-sqrt(2))/4)*2/sqrt(2)#

#A=9((sqrt(6)-sqrt(2))/(2sqrt(2)))#

#A=9(((sqrt(6)-sqrt(2))*sqrt(2))/(2sqrt(2)sqrt(2)))#

#A=9(sqrt(12)-2)/(4)#

#A=9/2(sqrt(3)-1)#

Now let's calculate #hat(AC)# and B

#hat(AC)=pi-(pi/12+3pi/4)=pi/6#

#B/sin hat(AC)=C/sin(hat (AB))#

#B/sin (pi/6)=9/sin(3pi/4))#

#B=(9*1/2)/(sqrt(2)/2)=9/sqrt(2)=9/2sqrt(2)#