Chapter 6.5, Problem 12E

To determine

To calculate: The complete triangle,

Answer to Problem 12E

The complete triangle is,

  $\begin{array}{c}\angle \text{A}={80}^{\circ }\\ \angle B={31}^{\circ }\\ \angle C={69}^{\circ }\end{array}$ and $\begin{array}{c}a=6.5\\ b=3.4\\ c=6.2\end{array}$ .

Explanation of Solution

Given information:

The figure,

Formula used:

Sum of interior angles of any triangle is ${180}^{\circ }$ . So, in any $\text{ΔABC}$ , $\angle \text{A}+\angle \text{B}+\angle \text{C}={180}^{\circ }$ .

According to law of sines, the lengths of the sides of the triangle are proportional to the sines of the corresponding opposite angles. In any triangle, $\Delta \text{XYZ}$

  $\frac{\text{sin X}}{\text{x}}=\frac{\text{sin Y}}{\text{y}}\text{=}\frac{\text{sin Z}}{\text{z}}$

Calculation:

Consider the given figure,

In the given figure, $\angle \text{A}={80}^{\circ }$ , $a=6.5$ and $b=3.4$ .

Recall that according to law of sines, the lengths of the sides of the triangle are proportional to the sines of the corresponding opposite angles. In any triangle, $\Delta \text{XYZ}$

  $\frac{\text{sin X}}{\text{x}}=\frac{\text{sin Y}}{\text{y}}\text{=}\frac{\text{sin Z}}{\text{z}}$

So, the value of $\angle \text{B}$ will be calculated as,

  $\begin{array}{c}\frac{\text{sin A}}{6.5}=\frac{\text{sin B}}{3.4}\\ \frac{\sin 80}{6.5}=\frac{\sin \text{B}}{3.4}\\ \sin \text{B}=\frac{12\sin 68}{12}\\ \text{B}={31}^{\circ }\end{array}$

Recall that the sum of interior angles of any triangle is ${180}^{\circ }$ . So, in any $\text{ΔABC}$ , $\angle \text{A}+\angle \text{B}+\angle \text{C}={180}^{\circ }$ .

So, $\angle \text{C}$ will be calculated as,

  $\begin{array}{c}\angle \text{A}+\angle \text{B}+\angle \text{C}={180}^{\circ }\\ {80}^{\circ }+{31}^{\circ }+\angle \text{C}={180}^{\circ }\\ \angle \text{C}={180}^{\circ }-{111}^{\circ }\\ \angle \text{C}={69}^{\circ }\end{array}$

Now, the value of $c$ will be calculated as,

  $\begin{array}{c}\frac{\text{sin A}}{6.5}=\frac{\text{sin C}}{c}\\ \frac{\sin 80}{6.5}=\frac{\sin 69}{c}\\ c=\frac{6.5\sin 69}{\sin 80}\\ c\approx 6.2\end{array}$

Thus, the complete triangle is,

  $\begin{array}{c}\angle \text{A}={80}^{\circ }\\ \angle B={31}^{\circ }\\ \angle C={69}^{\circ }\end{array}$ and $\begin{array}{c}a=6.5\\ b=3.4\\ c=6.2\end{array}$ .