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Arc CD is [tex][tex]$\frac{2}{3}$[/tex][/tex] of the circumference of a circle. What is the radian measure of the central angle?

A. [tex][tex]$\frac{2 \pi}{3}$[/tex][/tex] radians
B. [tex][tex]$\frac{3 \pi}{4}$[/tex][/tex] radians
C. [tex][tex]$\frac{4 \pi}{3}$[/tex][/tex] radians
D. [tex][tex]$\frac{3 \pi}{2}$[/tex][/tex] radians

Sagot :

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.