Chapter 9.4, Problem 22WE

To determine

To Find : The radius of the circle from point O.

Answer to Problem 22WE

The radius of the circle from point O is approx. 16.7 cm .

Explanation of Solution

Given information:

  

Consider the figure as below:

  

In the above figure circle contains chords $RS=20$ , point O is a center point of circle.

Measure of arc $RS=70°$

Suppose a line segment OM is a perpendicular segment from point O to chord RS and it bisects $\angle ROS$ into two equal parts which means $\angle ROM=\angle SOM$ .

In circle M is the midpoint of the chord RS which means $RS=RM+SM$ and $RM=SM$ .

it can be defined as:

  $RM=\frac{RS}{2}$

If $RS=20$ then:

  $\begin{array}{l}RM=\frac{RS}{2}\hfill \\ RM=\frac{\cancel{20}}{\cancel{2}}\hfill \\ RM=10\hfill \end{array}$

According to theorem , In the same circle if two minor arc are congruent only when their central angles are congruent.

In above figure is same circle if measure of arc $arcRS=70°$ then angle $\angle ROS=70$ because measure of minor arc and central angles are congruent and equal to each other.

Also a perpendicular segment OM which bisects $\angle ROS=70$ into two equal parts

  $\angle ROM=\angle SOM=35°$

Use trigonometric identity:

  $\sin \theta =\frac{perpendicular}{hypotenuse}$ in form right triangle ?OMR

Here perpendicular $RM=10$ , $hypotenuseOR=x$ and if $\angle ROM=\angle SOM=35°$ then

  $\theta =35°$

  $\begin{array}{l}sin35°=\frac{10}{x}\hfill \\ sin35°=\frac{10}{x}\hfill \end{array}$

Measure of $sin35°=0.573576436351$

  $\begin{array}{l}sin35°=\frac{10}{x}\hfill \\ 0.57357643651=\frac{10}{x}\hfill \end{array}$

Multiply both sides of equation $0.573576436351=\frac{10}{x}$ by x

  $\begin{array}{l}0.573576436351\times x=\frac{10}{\cancel{x}}\times \cancel{x}\hfill \\ 0.6x=10\hfill \end{array}$

Divide both sides of equation $0.6x=10$ by $0.6$

  $\begin{array}{l}\cancel{\frac{0.6x}{0.6}}\approx \cancel{\frac{100}{0.6}}\\ x\approx \frac{50}{3}\\ x\approx 16.666666667\\ x\approx 16.7\end{array}$

Hence, measure of radius $OR\approx 16.7$

Therefore, radius of the circle from point O is approx. 16.7 cm .