# A triangle has sides A, B, and C. The angle between sides A and B is (5pi)/12 and the angle between sides B and C is pi/12. If side B has a length of 9, what is the area of the triangle?

Oct 10, 2016

$10.8$

#### Explanation:

We can compute the third angle with the other two angles as the sum of the angles in a triangle is ${180}^{\circ}$

$\text{Third angle} = 180 - \left(75 + 15\right) = 180 - 90 = {90}^{\circ}$

As the triangle contains a $\text{right angle}$,we can use trigonometry

We need to find one more side to find the area of the triangle

So, we can use

color(orange)(tan(theta)=("opposite") /(" hypotenuse")

$\rightarrow \tan \left(a\right) = \frac{A}{9}$

$\rightarrow \tan \left(15\right) = \frac{A}{9}$

$\rightarrow 0.267 = \frac{A}{9}$

$\rightarrow A = 0.267 \cdot 9$

rArrcolor(green)(A=2.4

We can calculate the area of the triangle

color(blue)("Area of triangle"=1/2*h*b

Where,

color(red)(h="height"=2.4

color(red)(b=base=9

$\therefore \text{Area} = \frac{1}{2} \cdot 2.4 \cdot 9$

$\rightarrow \frac{1}{\cancel{2}} ^ 1 \cdot {\cancel{2.4}}^{1.2} \cdot 9$

$\rightarrow 1.2 \cdot 9$

rArrcolor(green)(10.8