Chapter 9, Problem 22RP

a.

To determine

To calculate: The measure ofarc BC

Answer to Problem 22RP

The measure of arc BC is $60°$ .

Explanation of Solution

Given information:

Circle with center P ,

  $m\angle CAB={30}^{\circ }$

Formula used:

  $m\stackrel{⌢}{BC}=2(m\angle CAB)$

Calculation:

  

The measure of arc BC is twice the measure of angle CAB .

  $m\stackrel{⌢}{BC}=2(m\angle CAB)=2({30}^{\circ })={60}^{\circ }$

b.

To determine

To calculate: The measure of arc AC .

Answer to Problem 22RP

The measure of arc ACis ${120}^{\circ }$ .

Explanation of Solution

Given information:

Circle with center P ,

  $m\angle CAB={30}^{\circ }$

Formula used:

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

Calculation:

  

The measure of arc AC and measure of arc BC form $180°$ .

  $m\stackrel{⌢}{AC}={180}^{\circ }-m\stackrel{⌢}{BC}={180}^{\circ }-{60}^{\circ }={120}^{\circ }$

c.

To determine

To find: The length of arcBC.

Answer to Problem 22RP

The length of arc BC is $6.28$ .

Explanation of Solution

Given information:

Circle with center P,

  $m\angle CAB={30}^{\circ }$

Formula used:

  $l(\stackrel{⌢}{BC)}=\frac{{60}^{\circ }}{{360}^{\circ }}(2\pi r)$

Calculation:

  

Point P is at (3,2) and Point B is at (9,2).

  $r=9-3=6$

  $r=6$

  $\begin{array}{l}l(\stackrel{⌢}{BC)}=\frac{{60}^{\circ }}{{360}^{\circ }}(2\pi r)\\ =l(\stackrel{⌢}{BC)}=\frac{{60}^{\circ }}{{360}^{\circ }}(2\pi \ast 6)\\ =2\pi \approx 6.28\end{array}$

d.

To determine

To find: The area of a shaded region.

Answer to Problem 22RP

The area of a shaded region is 18.85.

Explanation of Solution

Given information:

Circle with center P ,

  $m\angle CAB={30}^{\circ }$

Formula used:

  $A=\frac{{60}^{\circ }}{{360}^{\circ }}(\pi r^2)$

Calculation:

  

Point P is at (3,2) and Point B is at (9,2).

  $r=9-3=6$

  $r=6$

  $\begin{array}{l}A=\frac{{60}^{\circ }}{{360}^{\circ }}(\pi r^2)\\ $ =\frac{{60}^{\circ }}{{360}^{\circ }}(\pi {(6)}^2)$=6\pi \approx 18.85\end{array}$