# A triangle has sides A, B, and C. The angle between sides A and B is (5pi)/6 and the angle between sides B and C is pi/12. If side B has a length of 5, what is the area of the triangle?

Oct 22, 2017

$6.25$ $u n i t {s}^{2}$

#### Explanation:

We first need to calculate the the third angle.

$B = \pi - \left(\frac{\pi}{12} + \frac{5 \pi}{6}\right) = \frac{\pi}{12}$

We can calculate the length of side $a$ using the Sine Rule:

$\sin \frac{A}{a} = \sin \frac{B}{b} = \sin \frac{C}{c}$

$\sin \frac{\frac{\pi}{12}}{a} = \sin \frac{\frac{\pi}{12}}{5} \implies a = \frac{5 \sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{\pi}{12}\right)} = 5$

We can find the altitude using:

$a \cdot \sin \left(C\right)$

Since area is $\frac{1}{2} \times b a s e \times h e i g h t$

We have:

$\frac{1}{2} \cdot 5 \cdot 5 \cdot \sin \left(\frac{5 \pi}{6}\right) = 6.25$ $u n i t {s}^{2}$

Oct 23, 2017

color(magenta)(6.249 units^2 to the nearest 3 decimal places

#### Explanation:

$\therefore \frac{{\cancel{5 \pi}}^{\textcolor{m a \ge n t a}{5}}}{\cancel{6}} ^ \textcolor{m a \ge n t a}{1} \times {180}^{\textcolor{m a \ge n t a}{30}} / {\cancel{\pi}}^{\textcolor{m a \ge n t a}{1}} = {150}^{\circ} = \angle$ between sides A and B

$\therefore \frac{{\cancel{\pi}}^{\textcolor{m a \ge n t a}{1}}}{\cancel{12}} ^ \textcolor{m a \ge n t a}{1} \times {\cancel{180}}^{\textcolor{m a \ge n t a}{15}} / {\cancel{\pi}}^{\textcolor{m a \ge n t a}{1}} = {15}^{\circ} = \angle$ between sides B and C

$\therefore 180 - \left(150 + 15\right) = {15}^{\circ} = \angle$ between sides A and C

The $\triangle$= a isoceles triangle

Area of $\triangle =$ $\frac{1}{2}$ base$\times$ height

$\therefore$Base$= C = \frac{C}{\sin {150}^{\circ}} = \frac{5}{\sin {15}^{\circ}}$

multiply both sides by $\sin {150}^{\circ}$

$\therefore C = \frac{5 \times \sin {150}^{\circ}}{\sin {15}^{\circ}}$

$\therefore C = \frac{5 \times 0.5}{0.258819045}$

$\therefore C = \frac{2.5}{0.258819045}$

color(magenta)(C=9.659=Base

Perpendicular  heightcolor(magenta)(= D

$\therefore \frac{D}{5} = S \in {15}^{\circ}$

multiply both sides by $5$

$\therefore D = 5 \times \sin {15}^{\circ}$

$\therefore D = 5 \times 0.258819045$

:.color(magenta)(D=1.294 units perpendicular height

$\therefore$The area of the trianglecolor(magenta)(=1/2xx9.659xx1.294=6.249 units^2 to the nearest 3 decimal places