Question

consider circle o, in which arc xy measures 16π cm. the length of a radius of the circle is 32 cm. what is the circumference of the circle? what is the ratio of the arc length to the circumference? what is the measure of central angle xoy?

Explanation:

Step1: Calculate the circumference

The formula for the circumference of a circle is $C = 2\pi r$. Given $r = 32$ cm, so $C=2\pi\times32 = 64\pi$ cm.

Step2: Find the ratio of arc - length to circumference

The arc - length $l = 16\pi$ cm and $C = 64\pi$ cm. The ratio is $\frac{l}{C}=\frac{16\pi}{64\pi}=\frac{1}{4}$.

Step3: Calculate the central angle

The ratio of the central angle $\theta$ (in radians) to $2\pi$ is the same as the ratio of the arc - length to the circumference. Let the central angle be $\theta$. We know $\frac{\theta}{2\pi}=\frac{16\pi}{64\pi}$. Cross - multiplying gives $\theta=\frac{1}{4}\times2\pi=\frac{\pi}{2}$ radians. To convert to degrees, use the conversion factor $\frac{180^{\circ}}{\pi}$. So $\theta=\frac{\pi}{2}\times\frac{180^{\circ}}{\pi}=90^{\circ}$.

Answer:

Circumference: $64\pi$ cm
Ratio of arc length to circumference: $\frac{1}{4}$
Measure of central angle XOY: $90^{\circ}$ or $\frac{\pi}{2}$ radians