In a circle centered at point O, sector AOB has a radius of length r, a central angle of \… — GRE General Test

In a circle centered at point $O$ , sector $AOB$ has a radius of length $r$ , a central angle of $\theta$ degrees, an arc length of $L$ , an area of $A_s$ , and a perimeter of $P$ . If the ratio of the sector area to the arc length is $\frac{A_s}{L} = 4$ , and the ratio of the sector area to the sector perimeter is $\frac{A_s}{P} = \frac{4}{3}$ , which of the following statements must be true? Select all that apply.

The radius $r$ of the circle is $8$ .Answer
The area $A_s$ of sector $AOB$ is $32$ .Answer
C
The central angle $\theta$ is greater than $60^\circ$ .
The perimeter $P$ of sector $AOB$ is $24$ .Answer
E
The arc length $L$ of sector $AOB$ is $8\pi$ .

Answer

The true statements are that the radius of the circle is 8, the area of the sector is 32, and the perimeter of the sector is 24.

The ratio of sector area to arc length simplifies directly to half the radius,

\frac{r}{2} = 4

, giving a radius of

8

. Expressing the sector perimeter as the sum of the arc length and two radii (

L + 16

) and using

\frac{A_s}{P} = \frac{4L}{L + 16} = \frac{4}{3}

yields

L = 8

. From this, the sector area is

A_s = 32

and the sector perimeter is

P = 24

Step-by-Step Solution

Relate sector area

A_s

and arc length

L

to find the radius

r

A_s = \frac{\theta}{360}\pi r^2

and

L = \frac{\theta}{360}2\pi r

, so

\frac{A_s}{L} = \frac{r}{2} = 4 \implies r = 8

Dividing the sector area formula by the arc length formula cancels the central angle fraction and

\pi

Set up the ratio equation for

\frac{A_s}{P}

using

P = L + 2r

Since

r = 8

P = L + 16

. Given

A_s = 4L

, we have

\frac{4L}{L + 16} = \frac{4}{3} \implies 3L = L + 16 \implies 2L = 16 \implies L = 8

Expressing both sector area and perimeter in terms of arc length allows direct solution for

L

Calculate sector area

A_s

and perimeter

P

A_s = 4(8) = 32

and

P = 8 + 2(8) = 24

Substituting

L = 8

and

r = 8

gives the exact values for area and perimeter.

Determine the central angle

\theta

8 = \frac{\theta}{360} \cdot 2\pi(8) \implies 8 = \frac{16\pi \theta}{360} \implies \theta = \left(\frac{90}{\pi}\right)^\circ \approx 28.65^\circ

Plugging

L = 8

and

r = 8

into the arc length formula gives

\theta \approx 28.65^\circ

, which is less than

60^\circ

Key Concept

Relating sector area, arc length, and perimeter using proportional ratios and fundamental circle formulas.

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