Chapter 8, Problem 18CR

a.

To determine

To find: the measure of QN

Answer to Problem 18CR

The measure of QN is $=10\text{ units}$

Explanation of Solution

Given Information: Two right angled triangles with common altitude and some of the angles and one side being marked on the figure.

Formula used:

  

In a right triangle, the tangent of an angle is the ratio of opposite side to the adjacent side of the triangle expressed as,

  $\tan B=\frac{a}{b}$

Calculation: Consider the triangle given below.

  

Here, considering the triangle PQN ,

  $\begin{array}{c}\tan 32°=\frac{PN}{QN}\\ =\frac{PN}{x}\end{array}$ .

  $PN=x\tan 32°$ ......(1)

Now, considering the other right triangle PNR,

  $\begin{array}{l}\tan 32°=\frac{PN}{20-x}\\ PN=\left(20-x\right)\tan 32°\end{array}$ ......(2)

Equating (1) and (2),

  $\begin{array}{l}\left(20-x\right)\tan 32°=x\tan 32°\\ 20-x=x\\ 2x=20\\ x=10\end{array}$

Thus, measure of QN is 10 units.

b.

To determine

To find: the measure of PN

Answer to Problem 18CR

The measure of PN is $\simeq \text{6 units}$

Explanation of Solution

Given Information: Two right angled triangles with common altitude and some of the angles and one side being marked on the figure.

Formula used:

  

In a right triangle, the tangent of an angle is the ratio of opposite side to the adjacent side of the triangle expressed as,

  $\tan B=\frac{a}{b}$

Calculation: Consider the triangle given below.

  

Here, considering the triangle PQN ,

  $\tan 32°=\frac{PN}{QN}$ .

Substituting the measure of QN obtained in the previous part,

  $\begin{array}{c}PN=QN\tan 32°\\ =10\tan 32°\end{array}$

Using the table of trigonometric values in Page 311,

  $\begin{array}{c}PN=10\times 0.6249\\ =6.249\\ \simeq 6\end{array}$