In a circle centered at point O with a radius of 6 units, points P and Q lie on the circle… — GRE General Test

In a circle centered at point $O$ with a radius of $6$ units, points $P$ and $Q$ lie on the circle such that central angle $\angle POQ$ measures $120^\circ$ . Which of the following statements are true? Select all such statements.

The length of minor arc $PQ$ is $4\pi$ .Answer
The area of minor sector $POQ$ is $12\pi$ .Answer
C
The perimeter of sector $POQ$ is $4\pi$ .
D
The area of sector $POQ$ is $36\pi$ .
The length of chord $PQ$ is $6\sqrt{3}$ .Answer

Answer

The true statements are that the length of minor arc

PQ

4\pi

, the area of minor sector

POQ

12\pi

, and the length of chord

PQ

6\sqrt{3}

The arc length of a

120^\circ

central angle is

\frac{1}{3}

of the total circumference

12\pi

, which gives

4\pi

. The sector area is

\frac{1}{3}

of the total area

36\pi

, which gives

12\pi

. Furthermore, by geometry of an isosceles triangle with vertex angle

120^\circ

and congruent sides

6

, the base chord length is

2 \times 6 \sin(60^\circ) = 6\sqrt{3}

Step-by-Step Solution

Calculate the arc length of minor arc

PQ

Arc length =

4\pi

Arc length equals

\frac{\theta}{360^\circ} \times 2\pi r = \frac{120^\circ}{360^\circ} \times 2\pi(6) = 4\pi

Calculate the area of sector

POQ

Sector area =

12\pi

Sector area equals

\frac{\theta}{360^\circ} \times \pi r^2 = \frac{120^\circ}{360^\circ} \times \pi(6^2) = 12\pi

Determine the perimeter of sector

POQ

Perimeter =

4\pi + 12

The boundary of a sector consists of the curved arc length plus two straight radii:

4\pi + 6 + 6 = 4\pi + 12

Calculate the length of chord

PQ

Chord

PQ = 6\sqrt{3}

Using triangle

OPQ

with sides

OP = OQ = 6

and

\angle POQ = 120^\circ

, law of cosines or bisecting

\angle POQ

into two

30^\circ

60^\circ

90^\circ

triangles yields chord length

2 \times (6 \sin 60^\circ) = 2 \times 3\sqrt{3} = 6\sqrt{3}

Key Concept

Arc Length, Sector Area, and Chord Relationships in Circles

Estimated Time:1m 30s

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