Question

which statement correctly establishes the proportional relationship between the arc lengths and radii of circles o and r with the same central - angle measure?
\\(\frac{\text{length of }\overset{\frown}{np}}{on}=\frac{rq}{\text{length of }\overset{\frown}{qs}}\\)
\\(\frac{\text{length of }\overset{\frown}{np}}{rq}=\frac{\text{length of }\overset{\frown}{qs}}{on}\\)
\\(\frac{\text{length of }\overset{\frown}{np}}{\text{length of }\overset{\frown}{qs}}=\frac{on}{rq}\\)
\\(\frac{\text{length of }\overset{\frown}{np}}{on}=\frac{\text{length of }\overset{\frown}{qs}}{rq}\\)

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the central - angle in radians. If two circles have the same central - angle measure $\theta$, for the first circle with radius $r_1$ and arc - length $s_1$, and the second circle with radius $r_2$ and arc - length $s_2$, we have $s_1=r_1\theta$ and $s_2 = r_2\theta$. Then $\frac{s_1}{r_1}=\theta$ and $\frac{s_2}{r_2}=\theta$. Since $\theta$ is the same for both circles, we get $\frac{s_1}{r_1}=\frac{s_2}{r_2}$.

Step2: Identify arc - lengths and radii

Let the arc - length of circle $O$ be the length of $\overset{\frown}{NP}$ and its radius be $ON$, and the arc - length of circle $R$ be the length of $\overset{\frown}{QS}$ and its radius be $RQ$. Then the correct proportion is $\frac{\text{length of }\overset{\frown}{NP}}{ON}=\frac{\text{length of }\overset{\frown}{QS}}{RQ}$, which can be rewritten as $\frac{\text{length of }\overset{\frown}{NP}}{\text{length of }\overset{\frown}{QS}}=\frac{ON}{RQ}$.

Answer:

$\frac{\text{length of }\overset{\frown}{NP}}{\text{length of }\overset{\frown}{QS}}=\frac{ON}{RQ}$