Circle K has a radius of 10. Points P and Q lie on circle K such that central angle \angle… — GRE General Test

Circle $K$ has a radius of $10$ . Points $P$ and $Q$ lie on circle $K$ such that central angle $\angle POQ$ measures $72^\circ$ , where point $O$ is the center of circle $K$ . Which of the following statements are true? Select all that apply.

The length of minor arc $PQ$ is $4\pi$ .Answer
The area of sector $POQ$ is $20\pi$ .Answer
C
The perimeter of sector $POQ$ is $4\pi$ .
D
The area of sector $POQ$ is $40\pi$ .
E
The ratio of the area of sector $POQ$ to the total area of circle $K$ is $1$ to $4$ .

Answer

The correct statements are that the length of minor arc

PQ

4\pi

, and the area of sector

POQ

20\pi

The central angle of

72^\circ

corresponds to

\frac{72^\circ}{360^\circ} = \frac{1}{5}

of the entire circle. The circle has a circumference of

2\pi(10) = 20\pi

and an area of

\pi(10)^2 = 100\pi

. Taking

\frac{1}{5}

of the circumference gives an arc length of

4\pi

, and taking

\frac{1}{5}

of the total area gives a sector area of

20\pi

. Both of these statements are mathematically accurate.

Step-by-Step Solution

Find the central angle fraction relative to the full circle

\text{Fraction} = \frac{72^\circ}{360^\circ} = \frac{1}{5}

A complete circle subtends

360^\circ

, so the arc and sector comprise one-fifth of the circle.

Calculate the circumference and length of minor arc

PQ

\text{Circumference} = 2\pi(10) = 20\pi, \quad \text{Arc Length } PQ = \frac{1}{5} \times 20\pi = 4\pi

Arc length is the central angle fraction multiplied by the total circumference.

Calculate the total circle area and area of sector

POQ

\text{Total Area} = \pi(10)^2 = 100\pi, \quad \text{Sector Area } POQ = \frac{1}{5} \times 100\pi = 20\pi

Sector area is the central angle fraction multiplied by the total area of the circle.

Evaluate sector perimeter and area ratio for remaining options

\text{Perimeter} = 4\pi + 2(10) = 4\pi + 20; \quad \text{Ratio} = \frac{20\pi}{100\pi} = \frac{1}{5}

Sector perimeter requires adding the two straight radii to the arc length, and the ratio of sector area to total area is

1:5

Key Concept

Arc Length and Sector Area of a Circle

Estimated Time:1m 30s

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