In a circle centered at point O, the radius is 18 units. Points P and Q lie on the circle … — GRE General Test

In a circle centered at point $O$ , the radius is $18$ units. Points $P$ and $Q$ lie on the circle such that the area of sector $POQ$ is $54\pi$ square units. What is the ratio of the length of minor arc $PQ$ to the total perimeter of sector $POQ$ ?

$\frac{\pi}{\pi + 6}$ Answer
B
$\frac{1}{6}$
C
$\frac{\pi}{\pi + 3}$
D
$\frac{6\pi}{\pi + 6}$
E
$\frac{9\pi}{\pi + 6}$

Answer

The ratio of the length of minor arc

PQ

to the total perimeter of sector

POQ

\frac{\pi}{\pi + 6}

The full circle area is

324\pi

, making the sector

54\pi / 324\pi = 1/6

of the circle. The arc length is

1/6 \times 36\pi = 6\pi

. The sector perimeter is the arc length plus two radii (

6\pi + 36

). Taking the ratio of arc length to sector perimeter yields

6\pi / (6\pi + 36) = \pi / (\pi + 6)

Step-by-Step Solution

Calculate the total area of the circle and determine the fractional size of sector

POQ

Total area =

\pi(18)^2 = 324\pi

. Sector fraction =

\frac{54\pi}{324\pi} = \frac{1}{6}

Determining the fraction of the circle represented by the sector is required to find the arc length.

Calculate the length of minor arc

PQ

and the total perimeter of sector

POQ

Minor arc

PQ = \frac{1}{6} \times 2\pi(18) = 6\pi

. Sector perimeter =

6\pi + 2(18) = 6\pi + 36

The sector perimeter consists of the curved arc length plus the two straight radii

OP

and

OQ

Form and simplify the ratio of arc length to sector perimeter.

\frac{6\pi}{6\pi + 36} = \frac{6\pi}{6(\pi + 6)} = \frac{\pi}{\pi + 6}

Factoring out 6 from the numerator and denominator simplifies the expression to its lowest form.

Key Concept

Arc Length and Sector Perimeter Calculations

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