Chapter 12.7, Problem 2P

To determine

To determine the length of the shortest side.

Answer to Problem 2P

Shortest length is 20.1 in.

Explanation of Solution

Given information :

Two angles are given ${75}^{\circ },{51}^{\circ }$

Side opposite to ${75}^{\circ }$ is 25 in.

According to the properties of triangle, shortest side is always opposite to shortest angle and longest side is always opposite to largest angle.

Here third angle is,

  $\left({180}^{\circ }-{75}^{\circ }-{51}^{\circ }\right)={54}^{\circ }$

We have to find the side opposite to the angle ${51}^{\circ }$

Now according to law of sines,

In any triangle ABC,

  

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

Hence,

  $\begin{array}{l}\frac{25}{\sin {75}^{\circ }}=\frac{b}{\sin {51}^{\circ }}\\ b=\frac{\sin {51}^{\circ }\times 25}{\sin {75}^{\circ }}\\ b=20.1\end{array}$