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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