Question

this question has two parts. use the information to answer part a and part b. the three concentric circles with center p have the same central angle. part a which relationship is true? a. \\(\frac{\text{length }\widehat{ab}}{pa}<\frac{\text{length }\widehat{mn}}{pm}<\frac{\text{length }\widehat{xy}}{px}\\) b. \\(\frac{\text{length }\widehat{ab}}{pa}=\frac{\text{length }\widehat{mn}}{pm}=\frac{\text{length }\widehat{xy}}{px}\\) part b consider an arc with length s, subtended by central angle \\(\theta\\), in a circle with radius r. angle \\(\theta\\) is measured in radians. which statement is true? a. \\(\frac{s}{r}=\theta\\), where \\(\theta\\) increases as r increases b. \\(\frac{r}{s}=\theta\\), where \\(\theta\\) increases as r increases c. \\(\frac{s}{r}=\theta\\), where \\(\theta\\) does not change as r increases d. \\(\frac{r}{s}=\theta\\), where \\(\theta\\) does not change as r increases

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s$ of a circle with radius $r$ and central - angle $\theta$ (in radians) is $s = r\theta$. So, $\frac{s}{r}=\theta$.

Step2: Analyze Part A

For three concentric circles with the same central - angle $\theta$, if the arc - lengths are $s_1$, $s_2$, $s_3$ and the radii are $r_1$, $r_2$, $r_3$ respectively, from $s = r\theta$, we have $\frac{s_1}{r_1}=\frac{s_2}{r_2}=\frac{s_3}{r_3}=\theta$. In the given circles, if length $\widehat{AB}=s_1$, $PA = r_1$, length $\widehat{MN}=s_2$, $PM = r_2$, length $\widehat{XY}=s_3$, $PX = r_3$, then $\frac{\text{length }\widehat{AB}}{PA}=\frac{\text{length }\widehat{MN}}{PM}=\frac{\text{length }\widehat{XY}}{PX}=\theta$.

Step3: Analyze Part B

Since $\frac{s}{r}=\theta$ (where $\theta$ is the central - angle in radians), and $\theta$ is a constant for a given central - angle regardless of the value of the radius $r$ of the circle. As long as the central - angle $\theta$ is fixed, changing the radius $r$ does not change the value of $\theta$.

Answer:

Part A: B. $\frac{\text{length }\widehat{AB}}{PA}=\frac{\text{length }\widehat{MN}}{PM}=\frac{\text{length }\widehat{XY}}{PX}$
Part B: C. $\frac{s}{r}=\theta$, where $\theta$ does not change as $r$ increases