Chapter 11.4, Problem 3WE

To determine

To Find: Radius $r$ and apothem $a$ .

Answer to Problem 3WE

  $a=3.5,r=3.5\sqrt{2}$

Explanation of Solution

Given:

  

  

Here Given: $Area=49$

Solving

  $\begin{array}{l}m\angle AOB=\frac{{360}^{°}}{n}=\frac{360}{4}\\ m\angle AOB={90}^{°}\\ m\angle AOX=\frac{1}{2}m\angle AOB=\frac{1}{2}(90)={45}^{°}\end{array}$

Now,

  $\begin{array}{l}\tan \angle AOX=\tan {45}^{°}=\frac{AX}{OX}=\frac{s}{2(a)}=1\\ s=2a\end{array}$

Now

  $\begin{array}{l}\cos {45}^{°}=\frac{OX}{OA}=\frac{a}{r}=\frac{1}{\sqrt{2}}=0\\ r=a\sqrt{2}\end{array}$

Thus perimeter

  $p=4s$

Now area

  $\begin{array}{l}A=\frac{1}{2}ap\\ A=\frac{1}{2}\left(\frac{s}{2}\right)(4s)\\ A=s^2\end{array}$

  $\begin{array}{l}A=49\\ A=s\\ 49=(s)\\ s=7\\ i.e.\\ a=\frac{s}{2}=\frac{7}{2}\\ a=3.5\\ and\\ r=a\sqrt{2}=3.5\sqrt{2}\end{array}$