Chapter 12.7, Problem 1P

To determine

To determine the length of the shortest side.

Answer to Problem 1P

Shortest length is 29.3 cm.

Explanation of Solution

Given information :

Two angles are given ${32}^{\circ },{53}^{\circ }$

Longest side is 55 cm

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

Here longest angle is,

  $\left({180}^{\circ }-{32}^{\circ }-{53}^{\circ }\right)={95}^{\circ }$

We have to find the side opposite to this.

Now according to law of sines,

In any triangle ABC,

  

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

Hence,

  $\begin{array}{l}\frac{55}{\sin {95}^{\circ }}=\frac{b}{\sin {32}^{\circ }}\\ b=\frac{\sin {32}^{\circ }\times 55}{\sin {95}^{\circ }}\\ b=29.3\end{array}$