In a circle centered at point O, the radius is 6 units and central angle \angle POQ measur… — GRE General Test

In a circle centered at point $O$ , the radius is $6$ units and central angle $\angle POQ$ measures $120^\circ$ . Which of the following statements regarding sector $POQ$ and minor arc $PQ$ must be true? Select all such statements.

The length of minor arc $PQ$ is $4\pi$ units.Answer
The area of sector $POQ$ is $12\pi$ square units.Answer
C
The area of sector $POQ$ is $36\pi$ square units.
D
The length of minor arc $PQ$ is $12\pi$ units.
E
The perimeter of sector $POQ$ is $4\pi + 6$ units.

Answer

The correct statements are that the length of minor arc

PQ

4\pi

units and the area of sector

POQ

12\pi

square units.

The correct statements accurately apply the arc length and sector area formulas by multiplying the full circumference (

12\pi

) and full area (

36\pi

) by the central angle fraction

\frac{120^\circ}{360^\circ} = \frac{1}{3}

, yielding an arc length of

4\pi

units and a sector area of

12\pi

square units.

Step-by-Step Solution

Calculate the central angle fraction of the circle.

The fraction of the circle represented by sector

POQ

\frac{120^\circ}{360^\circ} = \frac{1}{3}

Arc length and sector area are proportional to the ratio of the central angle to

360^\circ

Calculate the arc length of minor arc

PQ

\text{Arc length} = \frac{1}{3} \times 2\pi(6) = 4\pi \text{ units}.

Arc length equals the central angle fraction times the total circumference

2\pi r

Calculate the area of sector

POQ

\text{Sector area} = \frac{1}{3} \times \pi(6^2) = 12\pi \text{ square units}.

Sector area equals the central angle fraction times the total circle area

\pi r^2

Evaluate the given statements against the calculated values.

Statements asserting an arc length of

4\pi

units and a sector area of

12\pi

square units are true. Other statements miscalculate by omitting the fraction or adding incorrect boundary components.

Comparing calculated values confirms the valid choices.

Key Concept

Arc Length and Sector Area Formulas

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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