Chapter 9.5, Problem 12CE

To determine

To calculate:the diameter from B and apply the case I

Explanation of Solution

Given information:

The measure of an inscribed angle is equal to half the measure of its intercepted arc.

Angle $\angle ABC$ inscribed in $○O$ where, point O lies outside $\angle ABC$ .

Consider the following figure,

  

Here, $OA=OB=OC$ equal to radius of the circle with center O,

Suppose, $\angle A=x$ this implies that $\angle ABO=x$ , as $OA=OB$ .

Therefore, $\angle AOE=2x$ Exterior angle of isosceles ΔAOB

  $\angle AOE=\angle EOC$ As BE bisects AC,

  $\begin{array}{l}\angle AOC=4x\hfill \\ m\stackrel{⌢}{AC}=4x\hfill \end{array}$

As EB bisects $\angle ABC$ ,

  $\begin{array}{l}\begin{array}{l}\angle ABC =\angle ABO+\angle OBC\hfill \\ =2x+2x\hfill \\ =4x\hfill \end{array}\\ =\frac{1}{2}m\stackrel{⌢}{AC}\end{array}$

The diameter from B is $=\frac{1}{2}m\stackrel{⌢}{AC}$