Chapter 8.4, Problem 10CE

To determine

To find: the length of AD and AC.

Answer to Problem 10CE

The length of AD and AC are $16$ and $8\sqrt{3}$ respectively.

Explanation of Solution

Given:

The hexagon is shown below:

The length of AB is 8.

Calculation:

Hexagonal is shown below:

For any regular hexagonal, triangles AOB, BOC and COD are equilateral triangles.

Hence,

  $AB=AO=CD=OD$

The Side AD is calculated as follows:

  $\begin{array}{c}AD=AO+OD\\ =AB+AB\\ =2\left(8\right)\\ =16\end{array}$

Now join the line AC and draw a perpendicular from B as shown below:

Side AP is calculated as follows:

  $\begin{array}{l}\sin 60°=\frac{AP}{AB}\\ \frac{\sqrt{3}}{2}=\frac{AP}{8}\\ AP=4\sqrt{3}\end{array}$

AC is twice of AP,

  $\begin{array}{c}AC=2AP\\ =2\left(4\sqrt{3}\right)\\ =8\sqrt{3}\end{array}$

Thus, the length of AD and AC are $16$ and $8\sqrt{3}$ respectively.