A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 5, respectively. The angle between A and C is #(11pi)/24# and the angle between B and C is # (7pi)/24#. What is the area of the triangle?

1 Answer
Alan P.
Feb 21, 2016

#"Area"_triangle=~~12.97#

Explanation:

Call the angle opposite side A as #/_a# (i.e. the angle between B and C);
and similarly the angle opposite Side B as #/_b#
and the angle opposite Side C as #/_c#

We are told (among other things) that:
#A=7, /_a=(7pi)/24, and /_c=(11pi)/24#

The Sine Law tells us
#color(white)("XXX")C/sin(c) = A/sin(a)#

So
#color(white)("XXX")C=7/(sin((7pi)/24))*sin((11pi)/24)#

Evaluating we get: #C=11#

We now have the lengths of the three sides and can apply Heron's Formula:
#color(white)("XXX")"Area"_triangle = sqrt(S(A-A)(S-B)(S-C))#
where #S# is the semi-perimeter (i.e. #(A+B+C)/2#)

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