Question

in each circle, a 50 - degree angle with a vertex at the center of the circle is drawn. how are minor arc lengths cd and ef related? the arc lengths are proportional: mcd = 4mef. they are the same because the central angle measure is the same. the arc lengths are proportional: mcd = 6mef. the arc lengths are proportional: mcd = 2mef.

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s = r\theta$ (where $\theta$ is the central - angle in radians and $r$ is the radius of the circle). If the central angles are equal ($\theta_1=\theta_2$), then $\frac{s_1}{s_2}=\frac{r_1}{r_2}$.
Let the radius of the first circle be $r_1$ and the radius of the second circle be $r_2$. Given that the central angle $\theta$ in both circles is $50^{\circ}$. First, convert the angle to radians: $\theta = 50\times\frac{\pi}{180}=\frac{5\pi}{18}$ radians.
The length of arc $CD$ in the first circle with radius $r_1$ is $s_1 = r_1\theta$, and the length of arc $EF$ in the second circle with radius $r_2$ is $s_2 = r_2\theta$.
If the radius of the circle with arc $CD$ is twice the radius of the circle with arc $EF$ (i.e., $r_1 = 2r_2$), then $s_1=r_1\theta$ and $s_2=r_2\theta$. Substituting $r_1 = 2r_2$ into the arc - length formula for $s_1$, we get $s_1=(2r_2)\theta = 2(r_2\theta)=2s_2$. So, $m\stackrel{\frown}{CD}=2m\stackrel{\frown}{EF}$ since the arc lengths are proportional to the radii when the central angles are equal.

Answer:

The arc lengths are proportional; $m\stackrel{\frown}{CD}=2m\stackrel{\frown}{EF}$