Arc CD is Two-thirds of the circumference of a circle. What is the radian measure of the central angle?

StartFraction 2 pi Over 3 EndFraction radians
StartFraction 3 pi Over 4 EndFraction radians
StartFraction 4 pi Over 3 EndFraction radians
StartFraction 3 pi Over 2 EndFraction radians

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olumideolawoyin olumideolawoyin · Answer 1 · 2024-06-17T10:07:23-04:00

Answer:

[tex]\frac{4}{3}\pi \: radians[/tex]

StartFraction 4 pi Over 3 EndFraction radians

Step-by-step explanation:

Arc CD is Two-thirds of the circumference of a circle.

[tex] \frac{central \: angle}{360} \times 2\pi \: r = \frac{2}{3} \times 2\pi \: r[/tex]

Divide both sides by 2πr

[tex] \frac{central \: angle}{360} = \frac{2}{3} [/tex]

In radians, since 360⁰ = 2π radians

[tex]\frac{central \: angle}{2\pi} = \frac{2}{3} [/tex]

3 × central angle = 2 × 2π

Divide both sides by 3

central angle = 4π / 3 rad

[tex]central \: angle = \frac{4}{3}\pi \: radians[/tex]

semsee45 semsee45 · Answer 2 · 2024-06-18T07:09:31-04:00

Answer:

[tex]\textsf{C)}\quad \dfrac{4\pi}{3}[/tex]

Step-by-step explanation:

The measure of an arc is equal to the measure of the central angle that subtends the arc.

Since angles around a point sum to 2π radians, the sum of the arcs of a circle is also 2π radians.

If arc CD is two-thirds of the circumference of a circle, then the measure of the central angle is two-thirds of 2π:

[tex]\dfrac{2}{3} \cdot 2\pi \\\\\\ \dfrac{4\pi}{3}[/tex]

So the radian measure of the central angle is:

[tex]\LARGE\boxed{\boxed{\dfrac{4\pi}{3}}}[/tex]