To find the length of arc ST in a circle where the chord ST is equal to the radius \( r \) of the circle, we need to follow a series of steps to understand the relationship between the chord, the radius, and the arc length. ### Step-by-Step Explanation 1. **Understanding the Circle and Chord**: - A circle is defined by its center and radius. The radius \( r \) is the distance from the center of the circle to any point on the circle. - A chord is a line segment whose endpoints lie on the circle. In this case, the chord ST has a length equal to the radius \( r \). 2. **Finding the Central Angle**: - To find the length of the arc ST, we first need to determine the central angle \( \theta \) that subtends the chord ST at the center of the circle. - We can use the relationship between the chord length, the radius, and the central angle. The formula for the length of a chord \( c \) in terms of the radius \( r \) and the central angle \( \theta \) (in radians) is: \[ c = 2r \sin\left(\frac{\theta}{2}\right) \] - Since the chord ST is equal to the radius \( r \), we set \( c = r \): \[ r = 2r \sin\left(\frac{\theta}{2}\right) \] - Dividing both sides by \( r \) (assuming \( r \neq 0 \)): \[ 1 = 2 \sin\left(\frac{\theta}{2}\right) \] - Solving for \( \sin\left(\frac{\theta}{2}\right) \): \[ \sin\left(\frac{\theta}{2}\right) = \frac{1}{2} \] - The angle whose sine is \( \frac{1}{2} \) is \( \frac{\pi}{6} \) radians. Therefore: \[ \frac{\theta}{2} = \frac{\pi}{6} \implies \theta = \frac{\pi}{3} \text{ radians} \] 3. **Calculating the Arc Length**: - The formula for the length of an arc \( L \) in a circle is given by: \[ L = r \theta \] - Substituting the values we have: \[ L = r \cdot \frac{\pi}{3} \] - Thus, the length of arc ST is: \[ L = \frac{\pi r}{3} \] ### Conclusion The correct answer is **D. \( \frac{\pi r}{3} \)**. ### Explanation of Other Options - **A. \( \frac{\pi r}{6} \)**: This option suggests a much smaller arc length, which would correspond to a smaller angle than \( \frac{\pi}{3} \). It does not match our calculated angle. - **B. \( \frac{\pi r}{2} \)**: This option implies a larger arc length, corresponding to a central angle of \( \frac{\pi}{2} \) radians, which is not applicable here since the chord length is equal to the radius. - **C. \( \frac{\pi r}{12} \)**: This option suggests an even smaller arc length, which again does not correspond to the angle we calculated. ### Revision Summary - The chord ST is equal to the radius \( r \) of the circle. - The central angle \( \theta \) that subtends the chord can be found using the sine function. - The length of the arc ST is calculated using the formula \( L = r \theta \). - The final answer for the length of arc ST is \( \frac{\pi r}{3} \).