Chapter 5.4, Problem 95E

To determine

To prove:Three squares of side $s$ are placed side by side as shown in given figure. Make a conjecture about the relationship between the sum $u+v$ and $w$ . Prove your conjecture by using the formulae for $\tan \left(u+v\right)$ .

Explanation of Solution

Given information:

The given figure as shown below.

  

Formula Used:

  $\tan \left(u+v\right)=\frac{\tan u+\tan v}{1-\tan u\tan v}$

Proof:

Let the figure as shown in fig. (1).

  

Fig. (1)

Using Pythagorean theorem in above fig. (1).

Using trigonometric ratio in $\Delta AEB$ .

  $\begin{array}{c}\tan u=\frac{\text{length}\text{of}\text{leg}\text{opposite}\angle u}{\text{length}\text{of}\text{leg}\text{adjacen}t\angle u}\\ =\frac{s}{3s}\\ =\frac{1}{3}\end{array}$

Using trigonometric ratio in $\Delta AEC$ .

  $\begin{array}{c}\tan v=\frac{s}{2s}\\ =\frac{1}{2}\end{array}$

Using trigonometric ratio in $\Delta AED$ .

  $\begin{array}{c}\tan w=\frac{s}{s}\\ =1\end{array}$

Drive the relationship between the sum $u+v$ and $w$ .

  $\tan \left(u+v\right)=\frac{\tan u+\tan v}{1-\tan u\tan v}$

Substitute the value of $\tan u$ and $\tan v$ .

  $\begin{array}{c}\tan \left(u+v\right)=\frac{\left(\frac{1}{3}\right)+\left(\frac{1}{2}\right)}{1-\left(\frac{1}{3}\right)\left(\frac{1}{2}\right)}\\ =\frac{5}{5}\\ =1\end{array}$

The result is equal to $w$ .

  $\begin{array}{c}\tan \left(u+v\right)=\tan w\\ u+v=w\end{array}$

From the aboveequation, it is clear that $u+v$ is always equal to $w$ for the above figure.

Hence, proved.