Chapter 9, Problem 19RP

a.

To determine

To calculate: The length of arcAF .

Answer to Problem 19RP

The length of arcAFis $120°$ .

Explanation of Solution

Given information:

A circle with center O,

  $\begin{array}{l}m\stackrel{⌢}{DE}=30°,\\ m\stackrel{⌢}{EB}=80°,\\ m\stackrel{⌢}{BF}={60}^{\circ }\end{array}$

Formula used:

The measure ofarc for half a circle is $180°$ and that of full circle is $360°$ .

Calculation:

  

The measure of arc for half a circle is $180°$ .

  $\begin{array}{l}m\stackrel{⌢}{AD}={180}^{\circ }-m\stackrel{⌢}{BE}-m\stackrel{⌢}{DE}\\ ={180}^{\circ }-{30}^{\circ }-{80}^{\circ }={70}^{\circ }\end{array}$

The measure of arc for full circle is $360°$ .

  $\begin{array}{l}m\stackrel{⌢}{AF}={360}^{\circ }-m\stackrel{⌢}{BF}-m\stackrel{⌢}{BE}-m\stackrel{⌢}{DE}-m\stackrel{⌢}{AD}\\ ={360}^{\circ }-{60}^{\circ }-{80}^{\circ }-{30}^{\circ }-{70}^{\circ }={120}^{\circ }\end{array}$

b.

To determine

To find: The measure of angleC .

Answer to Problem 19RP

The measure of angleCis $45°$ .

Explanation of Solution

Given information:

A circle with center O,

  $\begin{array}{l}m\stackrel{⌢}{DE}=30°,\\ m\stackrel{⌢}{EB}=80°,\\ m\stackrel{⌢}{BF}={60}^{\circ }.\end{array}$

Formula used:

The measure of secant-secant angle is one half the difference of its two intercepted arcs.

Calculation:

  

The measure of angle C is half the difference of arc AF and arc DE .

  $m\angle C=\frac{m\stackrel{⌢}{AF}-m\stackrel{⌢}{DE}}{2}=\frac{120°-30°}{2}=45°$

c.

To determine

To calculate: The measure of angleBAD .

Answer to Problem 19RP

The measure of angleBAD is $55°$ .

Explanation of Solution

Given information:

A circle with center O,

  $\begin{array}{l}m\stackrel{⌢}{DE}=30°,\\ m\stackrel{⌢}{EB}=80°,\\ m\stackrel{⌢}{BF}={60}^{\circ }\end{array}$

Formula used:

The measure of angle: $m\angle BAD=\frac{1}{2}(m(\stackrel{⌢}{BD})$ .

Calculation:

  

The measure of angle BAD is half the measure of arc BD .

  $\begin{array}{l}m\angle BAD=\frac{1}{2}(m(\stackrel{⌢}{BD})\\ =\frac{1}{2}({110}^{\circ })={55}^{\circ }\end{array}$