Chapter 12, Problem 30CR

a.

To determine

To find: The measure of arc $AD$ if $\overleftrightarrow{PA}$ and $\overleftrightarrow{PR}$ are tangent to circle with center O.

Answer to Problem 30CR

The measure of arc $AD$ is $98°$ .

Explanation of Solution

Given information:

  $\overleftrightarrow{PA}$ and $\overleftrightarrow{PR}$ are tangent to circle with center O.

Calculation:

  

  $\angle AED$ is supplementary to $\angle AEC$ .

  $\begin{array}{l}\angle AEC={118}^{\circ }\\ \angle AED={180}^{\circ }-{118}^{\circ }={62}^{\circ }\\ m\angle AED=\frac{1}{2}\left(\stackrel{⌢}{AD}+\stackrel{⌢}{CB}\right)\\ {62}^{\circ }=\frac{1}{2}\left(\stackrel{⌢}{AD}+{26}^{\circ }\right)\\ {62}^{\circ }=\frac{1}{2}\left(\stackrel{⌢}{AD}\right)+{13}^{\circ }\\ \frac{1}{2}\left(\stackrel{⌢}{AD}\right)={49}^{\circ }\\ \stackrel{⌢}{AD}={98}^{\circ }\end{array}$

b.

To determine

To calculate: The measure of angle Pif $\overleftrightarrow{PA}$ and $\overleftrightarrow{PR}$ are tangent to circle with center O.

Answer to Problem 30CR

The measure of angle Pis${82}^{\circ }$ .

Explanation of Solution

Given information:

  $\overleftrightarrow{PA}$ and $\overleftrightarrow{PR}$ are tangent to circle with center O.

Calculation:

  

  $\begin{array}{l}m\angle P=\frac{1}{2}\left(\stackrel{⌢}{ABD}+\stackrel{⌢}{AD}\right)\\ m\left(\stackrel{⌢}{ABD}\right)={360}^{\circ }-{98}^{\circ }={262}^{\circ }\\ m\angle P=\frac{1}{2}\left({262}^{\circ }-{98}^{\circ }\right)\\ m\angle P=\frac{1}{2}\left({164}^{\circ }\right)\\ m\angle P={82}^{\circ }\end{array}$