Chapter 9.6, Problem 31WE

To determine

To discover: and prove a relation between the measures of $arcCE$ and $arcBD$ .

Answer to Problem 31WE

The relation between measure of $CE$ and $BD$ is $2mBD=mCE$ .

Explanation of Solution

Given Information:

Given figure,

  

  $\overline{AC}$ and $\overline{AE}$ are secants of $\odot O$ .

Given $\overline{AB}\cong \overline{OB}$

Calculation:

Consider the given circle center $O$ , where $AC$ and $AE$ are secants of the circle.

And also consider that,

  $AB=OE$

Since $OE$ is the radius of circle and $OB$ is also radius of circle.

Therefore,

  $OB=OE$

Thus, it can be concluded that,

  $AB=OE$

In triangle $AOB$ ,

Since $AB=OE$ ,

Therefore,

  $\angle BAO=\angle BOA$

Now measure of $\angle BOA$ will be

  $\angle BOA=mBD$

Since, $AC$ and $AE$ are secants of circle,

Therefore,

  $\angle BAO=mCE-mBD$

As it is known that $\angle BAO=\angle BOA$ ,

So,

  $\begin{array}{c}\angle BAO=\angle BOA\\ mBD=mCE-mBD\end{array}$

Solve the equation to get,

  $\begin{array}{c}mBD=mCE-mBD\\ 2mBD=mCE\end{array}$

Hence, the relation between measure of $CE$ and $BD$ is $2mBD=mCE$ .