In circle O, points P and Q lie on the circumference such that the ratio of the minor arc … — GRE General Test

In circle $O$ , points $P$ and $Q$ lie on the circumference such that the ratio of the minor arc length $PQ$ to the radius $r$ of the circle is $\frac{5\pi}{6}$ . If the area of sector $POQ$ is $30\pi$ , what is the total perimeter of sector $POQ$ ?

$12\sqrt{2} + 5\sqrt{2}\pi$ Answer
B
$12 + 5\pi$
C
$5\sqrt{2}\pi$
D
$12 + 10\pi$
E
$12\sqrt{2} + 12\sqrt{2}\pi$

Answer

12\sqrt{2} + 5\sqrt{2}\pi

The central angle

\theta

in radians is equal to the ratio of minor arc length to radius, so

\theta = \frac{5\pi}{6}

. Using the area of a sector formula

A = \frac{1}{2}r^2\theta

, we substitute

A = 30\pi

to get

\frac{1}{2}r^2\left(\frac{5\pi}{6}\right) = 30\pi

, which simplifies to

r^2 = 72

, or

r = 6\sqrt{2}

. The arc length is then

s = r\theta = (6\sqrt{2})\left(\frac{5\pi}{6}\right) = 5\sqrt{2}\pi

. The total perimeter of the sector includes both bounding radii and the arc length:

2r + s = 12\sqrt{2} + 5\sqrt{2}\pi

Step-by-Step Solution

Relate the central angle in radians to the given ratio of arc length to radius.

The central angle

\theta

in radians is given by

\theta = \frac{\text{arc length}}{r} = \frac{5\pi}{6}

By definition of radian measure, arc length

s = r\theta

, so

\frac{s}{r} = \theta

Use the sector area formula to solve for the radius

r

Setting

\frac{1}{2}r^2\left(\frac{5\pi}{6}\right) = 30\pi

yields

\frac{5\pi}{12}r^2 = 30\pi \implies r^2 = 72 \implies r = 6\sqrt{2}

The area of a sector with central angle

\theta

(in radians) is

A = \frac{1}{2}r^2\theta

Calculate the minor arc length

s

Arc length

s = r\theta = (6\sqrt{2})\left(\frac{5\pi}{6}\right) = 5\sqrt{2}\pi

Multiplying the radius by the central angle in radians gives the length of the subtended arc.

Calculate the total perimeter of sector

POQ

Perimeter

= 2r + s = 2(6\sqrt{2}) + 5\sqrt{2}\pi = 12\sqrt{2} + 5\sqrt{2}\pi

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

Key Concept

Relationship between radian measure, sector area, arc length, and sector perimeter

Estimated Time:2m 0s

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