Answer
$65.94$ cm$^{2}$
Work Step by Step
First, we need to find the length of the side $c$ in order to use the formula for the Area of the triangle (SAS):
For this, we first find the angle $B$:
$A^{\circ}+B+C^{\circ}=180^{\circ}$
$30.5^{\circ}+B+112.60^{\circ}=180^{\circ}$
$143.1^{\circ}+B=180^{\circ}$
$B=180^{\circ}-143.1^{\circ}$
$B=36.9^{\circ}$
Now, we use the sine rule to find the length of the side $c$:
$\frac{b}{\sin B}=\frac{c}{\sin C}$
$\frac{13}{\sin 36.9}=\frac{c}{\sin 112.6}$
$c=\frac{13\times\sin 112.6}{\sin 36.9}$
$c=\frac{13\times 0.92321}{0.60042}$
$c=19.989$
The area of the triangle is half the product of the length of two sides and the sine of the angle included between them:
$Area=\frac{1}{2}bc \sin A$
We substitute the values of $A,b$ and $c$ in this formula and solve:
$Area=\frac{1}{2}bc \sin A$
$Area=\frac{1}{2}(13)(19.989) \sin 30.5^{\circ}$
$Area=\frac{1}{2}(259.86) \sin 30.5^{\circ}$
$Area=129.93\sin 30.5^{\circ}$
Using a calculator, $\sin 30.5^{\circ}=0.50754$. Therefore,
$Area=129.93\sin 30.5^{\circ}$
$Area=129.93(0.50754)$
$Area=65.94$
Therefore, the area of the triangle is $65.94$ cm$^{2}$.
