A triangle has sides A, B, and C. Sides A and B have lengths of 12 and 5, respectively. The angle between A and C is (5pi)/245π24 and the angle between B and C is (7pi)/247π24. What is the area of the triangle?

1 Answer
Lotusbluete
Jan 26, 2016

The area of the triangle is 30 " units"^230 units2.

Explanation:

Let the angle between AA and CC be betaβ, the angle between BB and CC be alphaα and finally, the angle between AA and BB be gammaγ.

We already know that

beta = (5 pi)/24 = 37.5^@β=5π24=37.5∘

and

alpha = (7 pi)/24 = 52.5^@α=7π24=52.5∘

We also know that the sum of the angles of the triangle must be 180^@ = pi180∘=π.

Thus, we can compute the third angle:

gamma = pi - (5pi)/24 - (7pi)/24 = pi - (12 pi)/24 = pi/2 = 90^@γ=π−5π24−7π24=π−12π24=π2=90∘

This means that the triangle has a right angle between AA and BB. This makes the calculation of the area easy:

"area" = 1/2 A * B * color(grey) (underbrace(sin(pi/2))_(=1)) = 1/2 * 12 * 5 = 30

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