Arc CD is $\frac{1}{4}$ of the circumference of a circle. What is the radian measure of the central angle?
A. $\frac{\pi}{4}$ radians
B. $\frac{\pi}{2}$ radians
C. $2 \pi$ radians
D. $4 \pi$ radians
Answer (2)
The arc CD is 4 1 of the circumference of a circle.
Calculate the arc length using the circumference formula: s = 4 1 ( 2 π r ) = 2 π r .
Apply the formula for the radian measure of a central angle: θ = r s .
Find the radian measure: θ = 2 π .
The radian measure of the central angle is 2 π .
Explanation
Problem Analysis Let's analyze the problem. We are given that arc CD is 4 1 of the circumference of a circle. We need to find the radian measure of the central angle corresponding to this arc.
Calculate Arc Length The circumference of a circle is given by 2 π r , where r is the radius of the circle. Since arc CD is 4 1 of the circumference, the length of arc CD is 4 1 ( 2 π r ) = 2 π r .
Radian Measure Formula The radian measure of a central angle, θ , is related to the arc length, s , and the radius, r , by the formula θ = r s . In our case, the arc length s is 2 π r .
Calculate Radian Measure Substituting the arc length into the formula, we get: θ = r 2 π r = 2 r π r = 2 π Thus, the radian measure of the central angle is 2 π radians.
Final Answer Therefore, the radian measure of the central angle is 2 π radians.
Examples
Imagine you're cutting a pizza into slices. If you cut the pizza into four equal slices, each slice represents 4 1 of the whole pizza. The angle of each slice at the center of the pizza is 2 π radians, which is 90 degrees. This is a practical example of how fractions of a circle relate to radian measures in everyday life.
The radian measure of the central angle corresponding to arc CD is 2 π radians, which represents 90 degrees. Therefore, the correct answer is A. 4 π radians. This is calculated based on the arc length being 4 1 of the total circumference of the circle.
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