Answer :
To determine the radian measure of the central angle corresponding to Arc CD, which is [tex]\(\frac{2}{3}\)[/tex] of the circumference of a circle, we can follow these steps:
1. Understand the Relationship Between Arc Length and Central Angle:
The circumference of a entire circle creates a central angle of [tex]\(2\pi\)[/tex] radians. Since Arc CD is given as [tex]\(\frac{2}{3}\)[/tex] of the full circumference, the central angle subtended by Arc CD will be [tex]\(\frac{2}{3}\)[/tex] of [tex]\(2\pi\)[/tex] radians.
2. Calculate the Central Angle:
[tex]\[ \text{Central Angle} = \frac{2}{3} \times 2\pi \][/tex]
3. Simplify the Expression:
[tex]\[ \text{Central Angle} = \frac{4\pi}{3} \][/tex]
4. Match the Result to the Given Choices:
Given the options, [tex]\(\frac{2\pi}{3}\)[/tex] radians, [tex]\(\frac{3\pi}{4}\)[/tex] radians, [tex]\(\frac{4\pi}{3}\)[/tex] radians, and [tex]\(\frac{3\pi}{2}\)[/tex] radians, the correct choice is:
[tex]\[ \frac{4\pi}{3} \text{ radians} \][/tex]
Therefore, the radian measure of the central angle is [tex]\(\frac{4\pi}{3}\)[/tex] radians, and this corresponds to the third option in the given choices.
1. Understand the Relationship Between Arc Length and Central Angle:
The circumference of a entire circle creates a central angle of [tex]\(2\pi\)[/tex] radians. Since Arc CD is given as [tex]\(\frac{2}{3}\)[/tex] of the full circumference, the central angle subtended by Arc CD will be [tex]\(\frac{2}{3}\)[/tex] of [tex]\(2\pi\)[/tex] radians.
2. Calculate the Central Angle:
[tex]\[ \text{Central Angle} = \frac{2}{3} \times 2\pi \][/tex]
3. Simplify the Expression:
[tex]\[ \text{Central Angle} = \frac{4\pi}{3} \][/tex]
4. Match the Result to the Given Choices:
Given the options, [tex]\(\frac{2\pi}{3}\)[/tex] radians, [tex]\(\frac{3\pi}{4}\)[/tex] radians, [tex]\(\frac{4\pi}{3}\)[/tex] radians, and [tex]\(\frac{3\pi}{2}\)[/tex] radians, the correct choice is:
[tex]\[ \frac{4\pi}{3} \text{ radians} \][/tex]
Therefore, the radian measure of the central angle is [tex]\(\frac{4\pi}{3}\)[/tex] radians, and this corresponds to the third option in the given choices.
