Question

below are circles c, f, and j. a central angle of 60° is shown in each. answer the following. when applicable, give the exact answer in terms of π in simplest form. (a) for each circle, find the length of the arc that is intercepted by the central angle. length of (widehat{ab}): □ cm length of (widehat{de}): □ cm length of (widehat{gh}): □ cm (b) for each circle, find the ratio of the arc length to the radius. (\frac{\text{length of }widehat{ab}}{\text{radius of circle }c}=square) (\frac{\text{length of }widehat{de}}{\text{radius of circle }f}=square) (\frac{\text{length of }widehat{gh}}{\text{radius of circle }j}=square) (c) the ratios suggest which of the following? the ratios increase by (\frac{pi}{3}) units as the radii increase by one unit. the length of the arc intercepted by a central angle is (\frac{pi}{3}) units more than the radius. the length of the arc intercepted by a central angle is proportional to the radius. the ratios increase by one unit as the lengths of the intercepted arcs increase by (\frac{pi}{3}) units. none of these.

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) intercepted by a central angle \(\theta\) (in radians) in a circle of radius \(r\) is \(s = r\theta\). First, convert the central angle \(\theta=60^{\circ}\) to radians. We know that \(\theta ( \text{in radians})=\frac{\pi}{180}\times\theta(\text{in degrees})\). So, \(\theta = 60\times\frac{\pi}{180}=\frac{\pi}{3}\) radians.

Step2: Calculate the arc - length for each circle

• For circle \(C\) with \(r = 3\) cm:

Using \(s=r\theta\), substitute \(r = 3\) and \(\theta=\frac{\pi}{3}\). Then \(s_{AB}=3\times\frac{\pi}{3}=\pi\) cm.

• For circle \(F\) with \(r = 1\) cm:

Using \(s=r\theta\), substitute \(r = 1\) and \(\theta=\frac{\pi}{3}\). Then \(s_{DE}=1\times\frac{\pi}{3}=\frac{\pi}{3}\) cm.

• For circle \(J\) with \(r = 2\) cm:

Using \(s=r\theta\), substitute \(r = 2\) and \(\theta=\frac{\pi}{3}\). Then \(s_{GH}=2\times\frac{\pi}{3}=\frac{2\pi}{3}\) cm.

Step3: Calculate the ratio of arc - length to radius for each circle

• For circle \(C\):

\(\frac{s_{AB}}{r_C}=\frac{\pi}{3}\).

• For circle \(F\):

\(\frac{s_{DE}}{r_F}=\frac{\frac{\pi}{3}}{1}=\frac{\pi}{3}\).

• For circle \(J\):

\(\frac{s_{GH}}{r_J}=\frac{\frac{2\pi}{3}}{2}=\frac{\pi}{3}\).

Answer:

(a)
Length of \(\overset{\frown}{AB}\): \(\pi\) cm
Length of \(\overset{\frown}{DE}\): \(\frac{\pi}{3}\) cm
Length of \(\overset{\frown}{GH}\): \(\frac{2\pi}{3}\) cm

(b)
\(\frac{\text{Length of }\overset{\frown}{AB}}{\text{Radius of circle }C}=\frac{\pi}{3}\)
\(\frac{\text{Length of }\overset{\frown}{DE}}{\text{Radius of circle }F}=\frac{\pi}{3}\)
\(\frac{\text{Length of }\overset{\frown}{GH}}{\text{Radius of circle }J}=\frac{\pi}{3}\)

(c)
The length of the arc intercepted by a central angle is proportional to the radius.