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 \( \frac{\pi m}{180} \). A. \( 2 \pi r \div \frac{360}{n} \) B. \( 2 \pi r \div \frac{180}{n} \) C. \( 2 \pi r \div \frac{270}{n} \) D. \( 2 \pi r \div \frac{90}{n} \)
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The expression that best completes Solomon's argument is \( 2 \pi r \div \frac{360}{n} \). This step connects the number of sectors formed by the central angle \( n^{\circ} \) to the circumference of the circle. By dividing the circumference \( 2 \pi r \) by the number of sectors, which is \( \frac{360}{n} \), you effectively find the arc length of one sector based on its central angle. In simpler terms, imagine you're slicing a pizza! If you know the total circumference and how many slices you have, simply divide the total length by the number of slices to find out how much crust each slice has. That's exactly what Solomon is doing with the circle and its arc length.
