Question

circle with center a, radius ac = 8 cm, central angle ∠cad = 50°, corresponding arc $overparen{cd}$; circle with center b, radius be = 2 cm, central angle ∠ebf = 50°, corresponding arc $overparen{ef}$.○ they are the same because the central angle measure is the same.○ the arc lengths are proportional: $overparen{cd}=2overparen{ef}$.○ the arc lengths are proportional: $overparen{cd}=4overparen{ef}$.○ the arc lengths are proportional: $overparen{cd}=6overparen{ef}$.

Explanation:

Step1: Recall arc length formula

The formula for arc length is $L = \frac{\theta}{360^\circ} \times 2\pi r$, where $\theta$ is the central angle, and $r$ is the radius of the circle.

Step2: Calculate arc length of $\overparen{CD}$

Substitute $\theta=50^\circ$, $r=8\ \text{cm}$:
$L_{CD} = \frac{50^\circ}{360^\circ} \times 2\pi \times 8 = \frac{80\pi}{36}$

Step3: Calculate arc length of $\overparen{EF}$

Substitute $\theta=50^\circ$, $r=2\ \text{cm}$:
$L_{EF} = \frac{50^\circ}{360^\circ} \times 2\pi \times 2 = \frac{20\pi}{36}$

Step4: Find the ratio of the arcs

Calculate $\frac{L_{CD}}{L_{EF}} = \frac{\frac{80\pi}{36}}{\frac{20\pi}{36}} = 4$, so $L_{CD}=4L_{EF}$.

Answer:

The arc lengths are proportional: $\overparen{CD} = 4\overparen{EF}$.