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$The circle has center O, and the measure of angle ROS is 150^(@). The length of minor arc widehat(RS) is what fraction of the circumference of the circle? (The number of degrees of arc in a circle is 360 .)$

The circle has center $O$ , and the measure of angle $\operatorname{ROS}$ is $150^{\circ}$ . The length of minor arc $\widehat{R S}$ is what fraction of the circumference of the circle? $\newline$ (The number of degrees of arc in a circle is $360$ .)

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Q. The circle has center

O

, and the measure of angle

\operatorname{ROS}

150^{\circ}

. The length of minor arc

\widehat{R S}

is what fraction of the circumference of the circle?

\newline

(The number of degrees of arc in a circle is

360

Understand Relationship: To solve this problem, we need to understand the relationship between the angle subtended by an arc at the center of a circle and the length of the arc itself. The length of an arc is proportional to the angle it subtends at the center of the circle. Since the circumference of the circle is the arc length corresponding to an angle of $360$ degrees, we can set up a ratio to find the fraction of the circumference that the minor arc $\widehat{RS}$ represents.
Calculate Fraction: The measure of angle $ROS$ is given as $150$ degrees. The total number of degrees in a circle is $360$ degrees. To find the fraction of the circumference that the arc $\widehat{RS}$ represents, we divide the angle measure of the arc by the total angle measure of the circle. $\newline$ Fraction of circumference = (Measure of angle $ROS$ ) / (Total degrees in a circle) $\newline$ Fraction of circumference = $\frac{150}{360}$
Simplify Fraction: Now we simplify the fraction $\frac{150}{360}$ to its simplest form. Both the numerator and the denominator are divisible by $10$ . $\newline$ Fraction of circumference = $\left(\frac{150}{10}\right) / \left(\frac{360}{10}\right)$ $\newline$ Fraction of circumference = $\frac{15}{36}$
Further Simplify Fraction: We can further simplify the fraction $\frac{15}{36}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $3$ . $\newline$ Fraction of circumference = $\left(\frac{15}{3}\right) / \left(\frac{36}{3}\right)$ $\newline$ Fraction of circumference = $\frac{5}{12}$