Chapter 13.4, Problem 6MRE

To determine

To find the length of the hexagon of side AD and AC.

Answer to Problem 6MRE

Length of the hexagon of side AD and AC are $\sqrt{3}x\text{ and }2x$ .

Explanation of Solution

Given information: Each side of a regular hexagon ABCDEF has length x.

Calculation: The required length of hexagon are obtained as,

Hexagon figure is,

  

In triangle ABC.

  

  $AB=BC=x$

  

From figure M is the midpoint of AC.

  $\begin{array}{l}AB=BC=x\\ AC=AM+MC=2AM\\ \cos 30=\frac{AM}{x}\\ AM=\frac{x\sqrt{3}}{2}\\ AC=2AM=x\sqrt{3}\\ AC=\sqrt{3}x\end{array}$

  

  $AD=2AO$ (O is the center of the hexagon).

Triangle AOB is equilateral triangle triangle.

  $\therefore AO=AB=x$

So,

  $\begin{array}{l}AD=2AO\\ AD=2x\end{array}$

Hence,length of the hexagon of side AD and AC are $\sqrt{3}x\text{ and }2x$ .