Sector S_1 belongs to a circle with radius r and has a central angle of measure \theta^\ci… — GRE General Test

Sector $S_1$ belongs to a circle with radius $r$ and has a central angle of measure $\theta^\circ$ , where $0 < \theta < 360$ . Sector $S_2$ belongs to a circle with radius $2r$ and has a central angle of measure $\left(\frac{\theta}{2}\right)^\circ$ . Which of the following statements must be true? Select all such statements.

The area of sector $S_2$ is twice the area of sector $S_1$ .Answer
The arc length of sector $S_2$ is equal to the arc length of sector $S_1$ .Answer
C
The perimeter of sector $S_2$ is twice the perimeter of sector $S_1$ .
The ratio of the area of sector $S_1$ to its arc length is half the ratio of the area of sector $S_2$ to its arc length.Answer
E
The perimeter of sector $S_2$ is equal to the perimeter of sector $S_1$ .

Answer

The true statements are that the area of sector

S_2

is twice the area of sector

S_1

, the arc length of sector

S_2

is equal to the arc length of sector

S_1

, and the ratio of area to arc length for sector

S_1

is half that of sector

S_2

The area of sector

S_2

is twice that of

S_1

because quadrupling

r^2

combined with halving the central angle results in a factor of 2. The arc length of sector

S_2

equals that of

S_1

because doubling

r

and halving the angle cancel each other out. The area-to-arc-length ratio of any sector reduces to

\frac{R}{2}

, so sector

S_1

with radius

r

has ratio

\frac{r}{2}

, which is half the ratio

r

of sector

S_2

Step-by-Step Solution

Calculate and compare the sector areas.

\text{Area}(S_1) = \frac{\theta}{360}\pi r^2

and

\text{Area}(S_2) = \frac{\theta/2}{360}\pi (2r)^2 = \frac{\theta}{720}\pi (4r^2) = \frac{2\theta}{360}\pi r^2 = 2 \cdot \text{Area}(S_1)

The sector area formula is

\text{Area} = \frac{\text{angle}}{360^\circ} \pi R^2

. Doubling the radius quadruples

R^2

, while halving the angle reduces the fraction by half, producing a net doubling of area.

Calculate and compare the arc lengths.

\text{Arc}(S_1) = \frac{\theta}{360}(2\pi r)

and

\text{Arc}(S_2) = \frac{\theta/2}{360}(2\pi \cdot 2r) = \frac{\theta}{360}(2\pi r) = \text{Arc}(S_1)

The arc length formula is

\text{Arc} = \frac{\text{angle}}{360^\circ} (2\pi R)

. Doubling the radius doubles

R

, while halving the angle halves the fraction, leaving the product unchanged.

Calculate and compare the sector perimeters.

\text{Perimeter}(S_1) = 2r + \text{Arc}(S_1)

and

\text{Perimeter}(S_2) = 4r + \text{Arc}(S_2) = 4r + \text{Arc}(S_1)

The perimeter of a sector consists of two straight radii and the curved arc length.

2 \cdot \text{Perimeter}(S_1) = 4r + 2\text{Arc}(S_1) \neq \text{Perimeter}(S_2)

Evaluate the area-to-arc-length ratios for both sectors.

For

S_1

\frac{\text{Area}(S_1)}{\text{Arc}(S_1)} = \frac{\frac{\theta}{360}\pi r^2}{\frac{\theta}{360}2\pi r} = \frac{r}{2}

. For

S_2

\frac{\text{Area}(S_2)}{\text{Arc}(S_2)} = \frac{2r}{2} = r

The ratio of area to arc length simplifies to

\frac{R}{2}

for any sector, so doubling the radius doubles this ratio.

Key Concept

Geometric properties of circle sectors, including proportional relationships between radius, central angle, arc length, sector area, and total sector perimeter.

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic