Chapter 8.6, Problem 22WE

To determine

To find: the length of diagonal $\overline{AC}$ .

Answer to Problem 22WE

The length of diagonal $AC$ is about 9 cm.

Explanation of Solution

Given Information:

Points $A,B$ and $C$ are three consecutive vertices of a regular decagon., whose sides are 16 cm long.

Calculation:

Consider the figure,

  

Three useful trigonometric ratios, given below in abbreviated form,

  $\begin{array}{l}\tan A=\frac{\text{opposite}}{\text{Adjacent}}\\ \sin A=\frac{\text{opposite}}{\text{Hypotenuse}}\\ \cos A=\frac{\text{Adjacent}}{\text{Hypotenuse}}\end{array}$

Angle in regular polygon of $n$ sides is given by,

  $\frac{360°}{n}=36°$

In case of diagram, number of sides is 10. Therefore, angle in regular decagon is given by

  $\frac{360°}{10}=36°$

Apply the sine formula in the right angle triangle for calculating length,

  $\sin C=\frac{AC}{16}$

Put the value of angle, to get

  $\sin 36°=\frac{AC}{16}$

Use cross-multiplication, to get

  $AC=16\sin 36°$

Use, the table of sine of angle provided, the approximate value of $AC$ is calculated as,

  $\begin{array}{l}AC=16\times 0.587\\ AC\approx 9\text{cm}\end{array}$

Therefore, the length of diagonal $AC$ is about 9 cm.