A triangle has sides A, B, and C. Sides A and B have lengths of 10 and 8, respectively. The angle between A and C is (13pi)/24 and the angle between B and C is (pi)24. What is the area of the triangle?

1 Answer
Dean R.
Apr 18, 2018

Since triangle angles add to pi we can figure out the angle between the given sides and the area formula gives
A = \frac 1 2 a b sin C = 10(sqrt{2} + sqrt{6}) .

Explanation:

It helps if we all stick to the convention of small letter sides a,b,c and capital letter opposing vertices A,B,C. Let's do that here.

The area of a triangle is A = 1/2 a b sin C where C is the angle between a and b.

We have B=\frac{ 13\pi}{24 } and (guessing it's a typo in the question) A=\pi/24.

Since triangle angles add up to 180^\circ aka \pi we get

C = \pi - \pi/24 - frac{13 pi}{24} = \frac{10 pi}{24} =\frac{5pi}{12}

\frac{5pi}{12} is 75^\circ. We get its sine with the sum angle formula:

sin 75^circ = sin(30 + 45) = sin 30 cos 45 + cos 30 sin 45

= (\frac 1 2 + frac sqrt{3} 2) \sqrt{2}/ 2

= \frac 1 4(sqrt(2) + sqrt(6))

So our area is

A = \frac 1 2 a b sin C = \frac 1 2 (10)(8) \frac 1 4(sqrt(2) + sqrt(6))

A = 10(sqrt{2} + sqrt{6})

Take the exact answer with a grain of salt because it's not clear we guessed correctly what the asker meant by the angle between B and C.

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