Chapter 8.5, Problem 24HP

To determine

To calculate:

The value of $x$

Answer to Problem 24HP

The value of $x$ is calculated as $7.9$

Explanation of Solution

Given information:

  

Calculation:

Firstly need to label the three sides of the big triangle $x,y+2,z$

. Then need to use Triangle Angle-Sum Theorem to find the missing angles of triangle:

Label the measures of the missing angles. Also, label the vertices as:

  

If is a $△ABC$ right triangle with acute $\angle A$ , then the tangent of $\angle A$ is the ratio of the length of

leg opposite $\angle A$ to the length of the leg adjacent $\angle A$ .

  $\begin{array}{l}\tan 15°=\frac{z}{GH}\\ GH=\frac{z}{\tan 15°}\\ GH+AH=\frac{z}{\tan 15°}\end{array}$

  $2+AH=\frac{z}{\tan 15°}$  ...... (1)

And:

  $\tan 20°=\frac{z}{AH}$

  $AH=\frac{z}{\tan 20°}$  ...... (2)

From eq. (1):

  $\begin{array}{l}2+\frac{z}{\tan 20°}=\frac{z}{\tan 15°}\\ 2=\frac{z}{\tan 15°}-\frac{z}{\tan 20°}\\ z=\frac{2}{\left(\frac{1}{\tan 15°}-\frac{1}{\tan 20°}\right)}\\ z\approx 2\end{array}$

Now, as per definition of tangent:

  $\begin{array}{l}\tan 70°=\frac{y}{z}\\ y=z\tan 70°\\ y=2\tan 70°\\ y\approx 5.58\end{array}$

Now, use the Pythagoras theorem to find value of $x$ :

  $\begin{array}{l}x=\sqrt{z^2+{\left(y+2\right)}^2}\\ x=\sqrt{2^2+{\left(5.58+2\right)}^2}\\ x=\sqrt{61.4564}\\ x\approx 7.9\end{array}$