Chapter 8, Problem 22CR

To determine

To find: the value of x

Answer to Problem 22CR

The value of x is $x=25.75$

Explanation of Solution

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

Formula used:

In a right triangle the sine of the angle is the ratio of the measure of side opposite to the angle to the hypotenuse.

  

It can be expressed as,

  $\sin B=\frac{a}{c}$

Similarly, 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 with acute angle 53⁰,

  $\tan 53°=\frac{x}{z}$ .

  $x=z\tan 53°$ ......(1)

Now, considering the other right triangle,

  $\begin{array}{c}\sin 76°=\frac{z}{20}\\ z=20\sin 76°\end{array}$

Referring to the table of trigonometric values in Page 311 in the textbook, the value of z can be obtained as,

  $\begin{array}{c}z=20\times 0.9703\\ =19.406\end{array}$

Now, substituting the value of z in (1) gives the value of x as,

  $x=19.406\tan 53°$

Again , with the use of the table of trigonometric values in Page 311 in the textbook, the value of x can be obtained as,

  $\begin{array}{c}x=19.406\tan 53°\\ =19.406\times 1.327\\ =25.75\\ \simeq 26\end{array}$