# A triangle has sides A, B, and C. The angle between sides A and B is (pi)/2 and the angle between sides B and C is pi/12. If side B has a length of 27, what is the area of the triangle?

Oct 22, 2016

$A r e a \triangle = 97.66$

#### Explanation:

Name the angles between sides$A \mathmr{and} B$ is
$\textcolor{red}{\angle M}$
Name the angles between sides$B \mathmr{and} C$ is
$\textcolor{b r o w n}{\angle N}$

GIVEN:
$\textcolor{red}{\angle M = \frac{\pi}{2}}$
$\textcolor{b r o w n}{\angle N = \frac{\pi}{12}}$
side$B = 27$

Let us find the area of this triangle:
The given $\triangle$ is right at M because $\textcolor{red}{\angle M = \frac{\pi}{2}}$

$\textcolor{b l u e}{A r e a = \frac{b \cdot h}{2}}$
so,
base $\textcolor{b l u e}{b} = B = 27$
height color(blue)h=color(green)A=???

$\textcolor{b l u e}{A r e a = \frac{27 \cdot \textcolor{g r e e n}{A}}{2}}$

Let us find side color(green)A=???
side $A$ is opposite to $\textcolor{b r o w n}{\angle N}$
Since given the adjacent side $B$ to the angle $\textcolor{b r o w n}{\angle N}$

$\textcolor{p u r p \le}{\tan N = \frac{A}{B}}$
$\tan \left(\frac{\pi}{12}\right) = \frac{A}{B}$
$\tan \left(\frac{\pi}{12}\right) = \frac{A}{27}$
$A = \tan \left(\frac{\pi}{12}\right) \cdot 27$

$\textcolor{g r e e n}{A = 7.235}$

Therefore,
$\textcolor{b l u e}{A r e a = \frac{27 \cdot \textcolor{g r e e n}{A}}{2}}$
$\textcolor{b l u e}{A r e a = \frac{27 \cdot \textcolor{g r e e n}{7.235}}{2}}$
$\textcolor{b l u e}{A r e a = 97.66}$