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:

Part (a)

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 \(\theta = 45^{\circ}\), so \(s=\frac{45}{360}\times2\pi r=\frac{\pi r}{4}\).

Step2: Calculate the arc - length for circle \(C\)

For circle \(C\) with \(r = 1\) mm. Substitute \(r = 1\) into the formula \(s=\frac{\pi r}{4}\). Then \(s=\frac{\pi\times1}{4}=\frac{\pi}{4}\) mm.

Step3: Calculate the arc - length for circle \(F\)

For circle \(F\) with \(r = 3\) mm. Substitute \(r = 3\) into the formula \(s=\frac{\pi r}{4}\). Then \(s=\frac{\pi\times3}{4}=\frac{3\pi}{4}\) mm.

Step4: Calculate the arc - length for circle \(J\)

For circle \(J\) with \(r = 2\) mm. Substitute \(r = 2\) into the formula \(s=\frac{\pi r}{4}\). Then \(s=\frac{\pi\times2}{4}=\frac{\pi}{2}\) mm.

Part (b)

Step1: Calculate the ratio for circle \(C\)

The radius of circle \(C\) is \(r = 1\) mm and the arc - length \(s=\frac{\pi}{4}\) mm. The ratio \(\frac{s}{r}=\frac{\frac{\pi}{4}}{1}=\frac{\pi}{4}\).

Step2: Calculate the ratio for circle \(F\)

The radius of circle \(F\) is \(r = 3\) mm and the arc - length \(s=\frac{3\pi}{4}\) mm. The ratio \(\frac{s}{r}=\frac{\frac{3\pi}{4}}{3}=\frac{\pi}{4}\).

Step3: Calculate the ratio for circle \(J\)

The radius of circle \(J\) is \(r = 2\) mm and the arc - length \(s=\frac{\pi}{2}\) mm. The ratio \(\frac{s}{r}=\frac{\frac{\pi}{2}}{2}=\frac{\pi}{4}\).

Part (c)

Since for all three circles \(\frac{\text{Arc length}}{\text{Radius}}=\frac{\pi}{4}\) (a constant), we know that the length of the arc intercepted by a central angle \(\theta\) (in this case \(\theta = 45^{\circ}\)) is proportional to the radius (\(s=\frac{\theta}{360}\times2\pi r\), and when \(\theta\) is constant, \(s\propto r\)).

Answer:

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