Question

below are circles c, f, and j. a central angle of 45° 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 \\( \overparen{a b} \\): \\( \square \mathrm{mm} \\) length of \\( \overparen{d e} \\): \\( \square \mathrm{mm} \\) length of \\( \overparen{g h} \\): \\( \square \mathrm{mm} \\) (b) for each circle, find the ratio of the arc length to the radius. \\( \frac{\text { length of } \overparen{a b}}{\text { radius of circle } c}=\square \\) \\( \frac{\text { length of } \overparen{d e}}{\text { radius of circle } f}=\square \\) \\( \frac{\text { length of } \overparen{g h}}{\text { radius of circle } j}=\square \\) (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 degrees) in a circle of radius \(r\) is \(s=\frac{\theta}{360}\times2\pi r\). Given \(\theta = 45^{\circ}\).

Step2: Calculate the length of \(\overset{\frown}{AB}\)

For circle \(C\) with \(r = 1\) mm.

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Step3: Calculate the length of \(\overset{\frown}{DE}\)

For circle \(F\) with \(r = 3\) mm.

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Step4: Calculate the length of \(\overset{\frown}{GH}\)

For circle \(J\) with \(r = 2\) mm.

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Step5: Calculate the ratios

• For circle \(C\): \(\frac{s_{AB}}{r_C}=\frac{\frac{\pi}{4}}{1}=\frac{\pi}{4}\)
• For circle \(F\): \(\frac{s_{DE}}{r_F}=\frac{\frac{3\pi}{4}}{3}=\frac{\pi}{4}\)
• For circle \(J\): \(\frac{s_{GH}}{r_J}=\frac{\frac{\pi}{2}}{2}=\frac{\pi}{4}\)

Answer:

(a)
Length of \(\overset{\frown}{AB}\): \(\frac{\pi}{4}\) mm
Length of \(\overset{\frown}{DE}\): \(\frac{3\pi}{4}\) mm
Length of \(\overset{\frown}{GH}\): \(\frac{\pi}{2}\) mm

(b)
\(\frac{\text{Length of }\overset{\frown}{AB}}{\text{Radius of circle }C}=\frac{\pi}{4}\)
\(\frac{\text{Length of }\overset{\frown}{DE}}{\text{Radius of circle }F}=\frac{\pi}{4}\)
\(\frac{\text{Length of }\overset{\frown}{GH}}{\text{Radius of circle }J}=\frac{\pi}{4}\)

(c) None of these. Since the ratio \(\frac{s}{r}=\frac{\theta}{180}\pi\) (when \(\theta\) is in degrees), and here \(\theta = 45^{\circ}\), \(\frac{s}{r}=\frac{\pi}{4}\) (a constant for the given central - angle measure, not following the statements in options A - D).