Chapter 57, Problem 1A

If $\angle A=30°$ and $\stackrel{⌢}{BD}=58°26\text{'}$ , find the measure of $\stackrel{⌢}{CD}$ .

To determine

The length of arc $\stackrel{⌢}{CD}.$

Answer to Problem 1A

Arc length $\stackrel{⌢}{CD}={118}^{°}26^{\prime }$

Explanation of Solution

Given:

  $\angle A={30}^{°}$

  $\stackrel{⌢}{BD}={58}^{°}26^{\prime }$

Concept used:

An angle formed at a point outside the circle by two secants, or two tangents is calculated by the given formula:

  $\angle A=\frac{1}{2}\left(\stackrel{⌢}{CD}-\stackrel{⌢}{BD}\right)$

Calculation:

The angle at point A is calculated by using the formula:

  $\angle A=\frac{1}{2}\left(\stackrel{⌢}{CD}-\stackrel{⌢}{BD}\right)$

  ${30}^{°}=\frac{1}{2}\left(\stackrel{⌢}{CD}-{58}^{°}26^{\prime }\right)$

  $\begin{array}{l}{30}^{°}\times 2=\stackrel{⌢}{CD}-{58}^{°}26^{\prime }\\ \stackrel{⌢}{CD}={30}^{°}\times 2+{58}^{°}26^{\prime }\\ \stackrel{⌢}{CD}={60}^{°}+{58}^{°}26^{\prime }\\ \stackrel{⌢}{CD}={118}^{°}26^{\prime }\end{array}$

Conclusion:

The measurement of the arc is $\stackrel{⌢}{CD}$ is ${118}^{°}26^{\prime }.$