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? π units 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.

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Step2: Find the ratio of arc length to circumference

Given arc length \(l=16\pi\) and \(C = 64\pi\). The ratio is \(\frac{l}{C}=\frac{16\pi}{64\pi}=\frac{1}{4}\)

Step3: Find the measure of central angle

The formula for arc length is \(l=\frac{\theta}{360^{\circ}}\times C\) (where \(\theta\) is the central angle). We know \(l = 16\pi\), \(C=64\pi\).

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Answer:

Circumference: \(64\pi\) units
Ratio: \(\frac{1}{4}\)
Measure of central angle \(X O Y\): \(90^{\circ}\)