Let's go through the solution step-by-step as if we're working this out carefully.
1. Understanding the Circumference:
- The circumference [tex]\( C \)[/tex] of a circle is given by [tex]\( C = \pi d \)[/tex], where [tex]\( d \)[/tex] is the diameter.
- Since the diameter [tex]\( d \)[/tex] is twice the radius [tex]\( r \)[/tex], we can write the circumference as [tex]\( C = 2 \pi r \)[/tex].
2. Dividing the Circle into Sectors:
- If we draw central angles each with a measure of [tex]\( n^\circ \)[/tex], the number of such sectors can be determined as [tex]\( \frac{360^\circ}{n^\circ} \)[/tex].
3. Arc Length of Each Sector:
- The arc length of a sector is the portion of the circumference that corresponds to a single central angle. For a circle divided into [tex]\( \frac{360^\circ}{n^\circ} \)[/tex] sectors, the arc length is the circumference divided by the number of such sectors.
- Therefore, the arc length [tex]\( L \)[/tex] for each sector is [tex]\( \frac{2 \pi r}{\frac{360^\circ}{n^\circ}} = 2 \pi r \cdot \frac{n^\circ}{360^\circ} \)[/tex].
4. Simplifying the Expression:
- The expression for the arc length can be simplified as:
[tex]\[
L = 2 \pi r \cdot \frac{n}{360}
\][/tex]
So, we see that the arc length of a sector with a central angle of [tex]\( n^\circ \)[/tex] is [tex]\( 2 \pi r \cdot \frac{n}{360} \)[/tex]. This can also be presented as:
[tex]\[
L = r \pi \cdot \frac{n}{180}
\][/tex]
By rechecking the provided options, the best fit to complete the argument is:
A. [tex]\( \frac{\pi n r}{180} \)[/tex].
Hence, each sector's arc length formula translates correctly to [tex]\( \frac{\pi n r}{180} \)[/tex] or equivalently [tex]\( 2 \pi r \cdot \frac{n}{360} \)[/tex].
