In a circle centered at point O, the radius is 12 units and central angle \angle AOB measu… — GRE General Test

In a circle centered at point $O$ , the radius is $12$ units and central angle $\angle AOB$ measures $150^\circ$ . Which of the following statements regarding sector $AOB$ and minor arc $AB$ are correct? Select all such statements.

The length of minor arc $AB$ is $10\pi$ units.Answer
The area of sector $AOB$ is $60\pi$ square units.Answer
The total perimeter of sector $AOB$ is $24 + 10\pi$ units.Answer
D
The length of minor arc $AB$ is $24\pi$ units.
E
The perimeter of sector $AOB$ is equal to $10\pi$ units.

Answer

The correct statements are those indicating that the minor arc length is

10\pi

units, the sector area is

60\pi

square units, and the perimeter of the sector is

24 + 10\pi

units.

The central angle fraction is

\frac{150^\circ}{360^\circ} = \frac{5}{12}

. Multiplying the full circumference (

24\pi

) by

\frac{5}{12}

gives an arc length of

10\pi

. Multiplying the full circle area (

144\pi

) by

\frac{5}{12}

gives a sector area of

60\pi

. Adding the two radii (

2 \times 12 = 24

) to the arc length (

10\pi

) gives the total sector perimeter of

24 + 10\pi

Step-by-Step Solution

Determine the central angle fraction of the circle.

The fraction is

\frac{150^\circ}{360^\circ} = \frac{5}{12}

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

360^\circ

Calculate the circumference of the circle and the arc length of minor arc

AB

Circumference =

2\pi(12) = 24\pi

. Arc length =

\frac{5}{12} \times 24\pi = 10\pi

units.

Arc length equals full circumference multiplied by the central angle fraction.

Calculate the total area of the circle and the area of sector

AOB

Circle Area =

\pi(12^2) = 144\pi

. Sector Area =

\frac{5}{12} \times 144\pi = 60\pi

square units.

Sector area equals total circle area multiplied by the central angle fraction.

Calculate the total perimeter of sector

AOB

Perimeter =

\text{arc length} + 2r = 10\pi + 2(12) = 24 + 10\pi

units.

The perimeter of a sector consists of the bounding arc plus the two radii.

Key Concept

Circles, Arc Lengths, and Sector Areas

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