Chapter 12.9, Problem 11WE

To determine

Find the value of $\left(a\right)$ of triangle ABC such that its area is $36\text{ sq units,}\angle B=30°\text{ and }c=8$ .

Answer to Problem 11WE

The value of $\left(a\right)$ of triangle ABC is $18.0\text{units}$ .

Explanation of Solution

Given information: The given triangle ABC has area $36\text{ sq units,}\angle B=30°\text{ and }c=8$ .

Calculation: Since the given triangle ABC has area $36\text{ sq units,}\angle B=30°\text{ and }c=8$ . And the given triangle has the case of side-angle-side where the given angle $\left(\angle B=30°\right)$ is included angle between the given two sides $a\text{ and }c$ . Therefore from the formula for area $\left(A\right)$ of triangle ABC the value for $\left(a\right)$ is evaluated as,

  $\begin{array}{c}A=\frac{1}{2}ac\sin B\\ 36=\frac{1}{2}\left(a\right)\left(8\right)\sin \left(30\right)\mathrm{....}\left\{\begin{array}{l}\Delta =36\text{,}\angle B=30°\\ \text{and }c=8\end{array}\right\}\\ \text{or, }a=\frac{\left(36\right)\left(2\right)}{\left(8\right)\sin \left(30\right)}\\ a=18.0\text{units}\end{array}$