Question

which proportion correctly defines θ?
a. $\frac{\text{length of }overline{pc}}{\text{length of }overline{pe}}=\frac{\text{length of }overline{hc}}{\text{length of }overline{de}}=\theta$
b. $\frac{\text{length of }overline{hc}}{\text{length of }overline{pc}}=\frac{\text{length of }overline{de}}{\text{length of }overline{pe}}=\theta$
c. $\frac{\text{length of }overline{pc}}{\text{length of }overline{hc}}=\frac{\text{length of }overline{pe}}{\text{length of }overline{de}}=\theta$
d. $\frac{\text{length of }overline{hc}}{\text{length of }overline{pe}}=\frac{\text{length of }overline{de}}{\text{length of }overline{pc}}=\theta$

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). For two concentric circles with central angle $\theta$ (the same for both circles), if the radius of the outer - circle is $r_1$ and the arc - length is $s_1$, and the radius of the inner - circle is $r_2$ and the arc - length is $s_2$, then $s_1=r_1\theta$ and $s_2 = r_2\theta$. So, $\frac{s_1}{r_1}=\frac{s_2}{r_2}=\theta$.

Step2: Identify radii and arcs

Let $\overline{PC}$ and $\overline{PE}$ be the radii of the outer and inner circles respectively, and $\widehat{BC}$ and $\widehat{DE}$ be the arcs of the outer and inner circles subtended by the central angle $\theta$. Then, by the arc - length formula, $\frac{\text{length of }\widehat{BC}}{\text{length of }\overline{PC}}=\frac{\text{length of }\widehat{DE}}{\text{length of }\overline{PE}}=\theta$.

Answer:

B. $\frac{\text{length of }\widehat{BC}}{\text{length of }\overline{PC}}=\frac{\text{length of }\widehat{DE}}{\text{length of }\overline{PE}}=\theta$