To determine the radian measure of the central angle corresponding to Arc CD, we need to first understand the relationship between the arc length and the central angle in a circle.
Step-by-step solution:
1. Understanding the given information:
Arc CD is [tex]\(\frac{2}{3}\)[/tex] of the circumference of the circle.
2. Recalling the full circumference:
The circumference of a circle is given by the formula [tex]\( C = 2\pi r \)[/tex], where [tex]\(r\)[/tex] is the radius of the circle.
3. Total radians in a circle:
In terms of radians, the entire circumference of a circle corresponds to [tex]\( 2\pi \)[/tex] radians. This is because a full circle is [tex]\(360^\circ\)[/tex], and [tex]\(360^\circ = 2\pi\)[/tex] radians.
4. Fraction of the circumference:
Since Arc CD is [tex]\(\frac{2}{3}\)[/tex] of the circumference, the corresponding central angle will be [tex]\(\frac{2}{3}\)[/tex] of the total radian measure of the circle.
5. Calculating the central angle:
To find the radian measure of the central angle, we multiply the total radian measure [tex]\(2\pi\)[/tex] by the fraction [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\text{Central angle} = \left(\frac{2}{3}\right) \times 2\pi
\][/tex]
6. Simplifying:
[tex]\[
\left(\frac{2}{3}\right) \times 2\pi = \frac{4\pi}{3}
\][/tex]
Therefore, the radian measure of the central angle corresponding to Arc CD is [tex]\(\frac{4 \pi}{3}\)[/tex] radians.
Therefore, the correct answer is:
[tex]\(\boxed{\frac{4 \pi}{3}}\)[/tex]
