Chapter 8, Problem 23RE

To determine

To find: The remaining angle(s) and side(s) of the triangle, if it(they) exist.

Answer to Problem 23RE

No triangleexists with the given measurements.

Explanation of Solution

Given:

The sides $a=2$ , $b=3$ and $c=1$ .

Calculation:

The Cosine Law for a triangle $\Delta ABC$ is given by the following equations:

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\left(\cos A\right)\\ b^2=a^2+c^2-2ac\left(\cos B\right)\\ c^2=a^2+b^2-2ab\left(\cos C\right)\end{array}$

Here, the capital letters represent the angles and the small letters represent the length of the respective sides of the triangle.

Given the 3 sides, all the angles can be determined as follows:

  $\begin{array}{c}{a}^2=b^2+c^2-2bc\left(\cos A\right)\\ 4=9+1-6\left(\cos A\right)\\ \cos A=\frac{6}{6}\\ A=0°\end{array}$

Since angle $A=0°$ , this implies that there is no triangle, only a straight line with the given measurements.