Question

solomon needs to justify the formula for the arc length of a sector. which expression best completes this argument?

• the circumference of a circle is given by the formula \\(c = \pi d\\), where \\(d\\) is the diameter.
• because the diameter is twice the radius, \\(c = 2\pi r\\).
• if equally sized central angles, each with a measure of \\(n^\circ\\), are drawn, the number of sectors that are formed will be equal to \\(\frac{360^\circ}{n^\circ}\\).
• the arc length of each sector is the circumference divided by the number of sectors, or ______.
• therefore, the arc length of a sector of a circle with a central angle of \\(n^\circ\\) is given by \\(2\pi r \cdot \frac{n}{360}\\) or \\(\frac{\pi r n}{180}\\).

a. \\(2\pi r \div \frac{180}{n}\\)
b. \\(2\pi r \div \frac{360}{n}\\)
c. \\(2\pi r \div \frac{270}{n}\\)
d. \\(2\pi r \div \frac{90}{n}\\)

Explanation:

Express the division of circumference by the number of sectors

$$ \text{Arc Length} = \frac{C}{\text{Number of sectors}} = \frac{2\pi r}{\frac{360}{n}} $$

Rewrite the division as multiplication by the reciprocal

$$ 2\pi r \div \frac{360}{n} = 2\pi r \cdot \frac{n}{360} $$

Match with the given options

$$ 2\pi r \div \frac{360}{n} $$

Answer:

• A. \(2\pi r \div \frac{180}{n}\)
• B. \(2\pi r \div \frac{360}{n}\) (Correct answer)
• C. \(2\pi r \div \frac{270}{n}\)
• D. \(2\pi r \div \frac{90}{n}\)