Q9. - Geometry

Question

a. Draw a circle. Place points A,B, and C on it in such positions that 

                            \$[m\stackrel{⏜}{AB}+m\stackrel{⏜}{BC}\mathrm{\backslash ]}$ does not equal 

 b. Does your example in part (a) contradict Postulate $16$?

Step-by-Step Solution

Answer

(a) The final answer is $50°$ and $310°$ .

(b) The final answer  is $NO$.

a. Step 1. Given information:

The given information is about the circle where $m\stackrel{⏜}{AB}$ and $m\stackrel{⏜}{BC}$  are the arcs of the circle.               

Step 2. Concept used:

$m\stackrel{⏜}{AB}$ means central angle of arc $AB$ and $m\stackrel{⏜}{BC}$ means central angle of arc $BC.$ .

   In the figure shown below, it is clear that $m\stackrel{⏜}{AB}+m\stackrel{⏜}{BC}$ does not equal to $m\stackrel{⏜}{AC}.$

Step 3. Applying the concept:

$mAB$ means central angle of arc $AB$ and $m\stackrel{⏜}{BC}$means central angle of arc $BC.$.

   In the figure shown below, it is clear that $m\stackrel{⏜}{AB}+m\stackrel{⏜}{BC}$ does not equal to $m\stackrel{⏜}{AC}.$

Here, $\begin{array}{c}m\stackrel{⏜}{AB}=90+110\\ =200\end{array}$

$\begin{array}{l}m\stackrel{⏜}{BC}=110\\ m\stackrel{⏜}{AC}=90\end{array}$

b. Step 1. Given information:

The given information is about the circle where $m\stackrel{⏜}{AB}$ and $m\stackrel{⏜}{BC}$  are the arcs of the circle

Step 2. Concept used:

 $m\stackrel{⏜}{AB}$  means central angle of arc $AB$and $m\stackrel{⏜}{BC}$ means central angle of arc $BC.$ .

   In the figure shown below, it is clear that $m\stackrel{⏜}{AB}+m\stackrel{⏜}{BC}$ does not equal to $m\stackrel{⏜}{AC}.$

Step 3. Applying the concept:

 $m\stackrel{⏜}{AB}$  means central angle of arc $AB$and $m\stackrel{⏜}{BC}$ means central angle of arc $BC.$ .

   In the figure shown below, it is clear that $m\stackrel{⏜}{AB}+m\stackrel{⏜}{BC}$ does not equal to $m\stackrel{⏜}{AC}.$

Here, $\begin{array}{c}m\stackrel{⏜}{AB}=90+110\\ =200\end{array}$

 $\begin{array}{l}m\stackrel{⏜}{BC}=110\\ m\stackrel{⏜}{AC}=90\end{array}$

According to theorem,

$m\stackrel{⏜}{AC}+m\stackrel{⏜}{BC}=m\stackrel{⏜}{AB}$

It does not contradict the postulate, here points are interchanged.