Arc CD is $\frac{2}{3}$ of the circumference of a circle. What is the radian measure of the central angle?
A. $\frac{2 \pi}{3}$ radians
B. $\frac{3 \pi}{4}$ radians
C. $\frac{4 \pi}{3}$ radians
D. $\frac{3 \pi}{2}$ radians
Answer (2)
Recognize that the full circumference of a circle corresponds to 2 π radians.
Multiply the fraction of the circumference, 3 2 , by 2 π to find the central angle in radians.
Perform the calculation: 3 2 × 2 π = 3 4 π .
State the final answer: 3 4 π
Explanation
Problem Analysis We are given that arc CD is 3 2 of the circumference of a circle. We need to find the radian measure of the central angle corresponding to this arc.
Solution Plan The circumference of a circle corresponds to a central angle of 2 π radians. Since arc CD is 3 2 of the circumference, we need to multiply 3 2 by 2 π to find the radian measure of the central angle.
Calculation Now, let's calculate the radian measure of the central angle: 3 2 × 2 π = 3 4 π So, the radian measure of the central angle is 3 4 π radians.
Final Answer Therefore, the radian measure of the central angle is 3 4 π radians.
Examples
Imagine you're baking a pie and you cut a slice that's 3 2 of the whole pie. The angle of that slice, measured from the center, would be 3 4 π radians. This concept is useful in many real-world applications, such as calculating distances on a circular track or designing gears.
The radian measure of the central angle corresponding to arc CD, which is 3 2 of the circumference of a circle, is 3 4 π radians. Thus, the correct answer is C . 3 4 π radians.
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