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 $(5pi)/24$ and the angle between B and C is $(7pi)/24$ . What is the area of the triangle?

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Answer 1 · 2016-01-26T08:26:44

The area of the triangle is $30 " units"^2$ .

Let the angle between $A$ and $C$ be $beta$ , the angle between $B$ and $C$ be $alpha$ and finally, the angle between $A$ and $B$ be $gamma$ .

We already know that

$beta = (5 pi)/24 = 37.5^@$

and

$alpha = (7 pi)/24 = 52.5^@$

We also know that the sum of the angles of the triangle must be $180^@ = pi$ .

Thus, we can compute the third angle:

$gamma = pi - (5pi)/24 - (7pi)/24 = pi - (12 pi)/24 = pi/2 = 90^@$

This means that the triangle has a right angle between $A$ and $B$ . This makes the calculation of the area easy:

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

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)/24(5pi)/24 and the angle between B and C is (7pi)/24 (7pi)/24. What is the area of the triangle?