Chapter 9.4, Problem 2DC

Solution

To find: The conjecture how the ratio of side length of any triangle is related to the opposite angles of any triangle.

Given information:

Change the shape of the triangle by dragging its vertices and observe how the ratio we found changes. Make a conjecture about how these ratios are related for any triangle.

  

Explanation:

From the given triangle, it is found that:

  $\frac{\sin \text{}A}{a}=\frac{\sin \text{}B}{b}=\frac{\sin \text{}\text{}\text{}C}{c}=0.28$

Now drag the vertices of triangle $ABC$ . Let following be the sides and the angles formed:

  $\begin{array}{l}a=25,\text{}b=27.7\text{and}c=16.2\\ A={63}^{\circ },B={82}^{\circ }\text{and}C={35}^{\circ }\end{array}$

Then, find the ratios as follows:

  $\frac{\sin \text{}{63}^{\circ }}{25}=\frac{\sin \text{}{82}^{\circ }}{27.7}=\frac{\sin \text{}\text{}\text{}{35}^{\circ }}{16.2}=0.035$

As we drag the vertices of the triangle, we will see that although the ratio changes it is the same for the three sides of the triangle.

Hence, the conjecture is:

The Law of Sines relates the ratio of side lengths of any triangle to their respective opposite angles. It is given as:

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