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Geometry and Trigonometry / Circles Difficulty: Hard

Point $O$ is the center of a circle. The measure of arc $R S$ on this circle is $100 degree$ . What is the measure, in degrees, of its associated angle $R O S$ ?

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Explanation

The correct answer is $100$ . It's given that point $O$ is the center of a circle and the measure of arc $R S$ on the circle is $100 degree$ . It follows that points $R$ and $S$ lie on the circle. Therefore, $\bar{O R}$ and $\bar{O S}$ are radii of the circle. A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle. Therefore, $∠ R O S$ is a central angle. Because the degree measure of an arc is equal to the measure of its associated central angle, it follows that the measure, in degrees, of $∠ R O S$ is $100$ .