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Difficulty: EasyCircles, Arc Lengths, and Sector Areas

A circle has a radius of 1010 units. A sector within this circle is defined by a central angle of 72∘72^\circ. Which of the following statements about this sector are true? Select all such statements.

  1. The arc length of the sector is 4π4\pi units.Answer
  2. The area of the sector is 20π20\pi square units.Answer
  3. C
    The area of the sector is 100π100\pi square units.
  4. D
    The arc length of the sector is 20π20\pi units.
  5. The ratio of the sector's area to the circle's total area is 11 to 55.Answer

Answer

The statements confirming an arc length of 4π4\pi units, a sector area of 20π20\pi square units, and a sector-to-total area ratio of 11 to 55 are correct.
The central angle fraction is 72∘360∘=15\frac{72^\circ}{360^\circ} = \frac{1}{5}. Multiplying the full circumference 20π20\pi by 15\frac{1}{5} gives an arc length of 4π4\pi. Multiplying the total area 100π100\pi by 15\frac{1}{5} gives a sector area of 20π20\pi. The ratio of sector area to total area is also equal to 15\frac{1}{5}.

Step-by-Step Solution

1
Find the central angle fraction of the circle
The fraction is 72∘360∘=15\frac{72^\circ}{360^\circ} = \frac{1}{5}
Arc length and sector area are proportional to the central angle relative to a full 360∘360^\circ turn.
2
Calculate the arc length of the sector
Arc Length =15×2π(10)=4π= \frac{1}{5} \times 2\pi(10) = 4\pi units
The arc length is the circle's circumference multiplied by the central angle fraction.
3
Calculate the area of the sector
Sector Area =15×π(10)2=20π= \frac{1}{5} \times \pi(10)^2 = 20\pi square units
The sector area is the circle's total area multiplied by the central angle fraction.
4
Determine the area ratio
Ratio =Sector AreaTotal Area=20π100π=15= \frac{\text{Sector Area}}{\text{Total Area}} = \frac{20\pi}{100\pi} = \frac{1}{5}
The ratio of the sector area to the total area is identical to the central angle fraction.

Key Concept

Arc Length and Sector Area Formulas
Estimated Time:1m 0s
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Topics

Circles, Arc Lengths, and Sector Areas

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