Question

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 (overline{ab}): (\frac{pi}{4}) mm length of (overline{de}): (\frac{3pi}{4}) mm length of (overline{gh}): (\frac{pi}{2}) mm (b) for each circle, find the ratio of the arc length to the radius. length of (overline{ab}): radius of circle (c=\frac{pi}{4}) length of (overline{de}): radius of circle (f=\frac{pi}{4}) length of (overline{gh}): radius of circle (j=\frac{pi}{4}) (c) the ratios suggest which of the following? 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}{4}) units. the length of the arc intercepted by a central angle is (\frac{pi}{4}) units more than the radius. the ratios increase by (\frac{pi}{4}) units as the radii increase by one unit. 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 \(s = r\theta\). Given \(\theta=45^{\circ}=\frac{\pi}{4}\) radians.

Step2: Calculate the length of arc \(AB\)

For circle \(C\) with \(r = 1\) mm. Using \(s=r\theta\), substitute \(r = 1\) and \(\theta=\frac{\pi}{4}\). So, \(s_{AB}=1\times\frac{\pi}{4}=\frac{\pi}{4}\) mm.

Step3: Calculate the length of arc \(DE\)

For circle \(F\) with \(r = 3\) mm. Using \(s=r\theta\), substitute \(r = 3\) and \(\theta=\frac{\pi}{4}\). So, \(s_{DE}=3\times\frac{\pi}{4}=\frac{3\pi}{4}\) mm.

Step4: Calculate the length of arc \(GH\)

For circle \(J\) with \(r = 2\) mm. Using \(s=r\theta\), substitute \(r = 2\) and \(\theta=\frac{\pi}{4}\). So, \(s_{GH}=2\times\frac{\pi}{4}=\frac{\pi}{2}\) mm.

Step5: Find the ratio of arc - length to radius

For a general circle with radius \(r\) and arc - length \(s=r\theta\) (\(\theta=\frac{\pi}{4}\) radians), the ratio \(\frac{s}{r}=\theta\). Since \(\theta=\frac{\pi}{4}\) (constant), when we consider \(\frac{\text{Length of }AB}{\text{Radius of }C}=\frac{\frac{\pi}{4}}{1}=\frac{\pi}{4}\), \(\frac{\text{Length of }DE}{\text{Radius of }F}=\frac{\frac{3\pi}{4}}{3}=\frac{\pi}{4}\), \(\frac{\text{Length of }GH}{\text{Radius of }J}=\frac{\frac{\pi}{2}}{2}=\frac{\pi}{4}\)

Answer:

(a) Length of \(AB:\frac{\pi}{4}\) mm; Length of \(DE:\frac{3\pi}{4}\) mm; Length of \(GH:\frac{\pi}{2}\) mm.
(b) \(\frac{\text{Length of }AB}{\text{Radius of }C}=\frac{\pi}{4}\); \(\frac{\text{Length of }DE}{\text{Radius of }F}=\frac{\pi}{4}\); \(\frac{\text{Length of }GH}{\text{Radius of }J}=\frac{\pi}{4}\)
(c) The ratios suggest that the length of the arc intercepted by a central angle is proportional to the radius.