Question

below are circles ( c ), ( f ), and ( j ).
a central angle of ( 150^{circ} ) is shown in each.

answer the following.
when applicable, give the exact answer in terms of ( pi ) 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{m} )

length of ( overparen{d e} ): ( square mathrm{m} )

length of ( overparen{g h} ): ( square mathrm{m} )

(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?

( \bigcirc ) the ratios increase by ( \frac{5 pi}{6} ) units as the radii increase by one unit.

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

( \bigcirc ) the ratios increase by one unit as the lengths of the intercepted
arcs increase by ( \frac{5 pi}{6} ) units.

( \bigcirc ) the length of the arc intercepted by a central angle is ( \frac{5 pi}{6} ) units
more than the radius.

( \bigcirc ) 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 \(s=\frac{\theta}{360}\times2\pi r\). Here \(\theta = 150^{\circ}\).
For circle \(C\) with \(r = 2\)m:

$$
LATEXBLOCK0
$$

Step2: For circle \(F\) with \(r = 3\)m

$$
LATEXBLOCK1
$$

Step3: For circle \(J\) with \(r = 1\)m

$$
LATEXBLOCK2
$$

Step4: Calculate the ratios

For circle \(C\): \(\frac{s_{AB}}{r}=\frac{\frac{5\pi}{3}}{2}=\frac{5\pi}{6}\)
For circle \(F\): \(\frac{s_{DE}}{r}=\frac{\frac{5\pi}{2}}{3}=\frac{5\pi}{6}\)
For circle \(J\): \(\frac{s_{GH}}{r}=\frac{\frac{5\pi}{6}}{1}=\frac{5\pi}{6}\)

Answer:

(a) Length of \(\overset{\frown}{AB}\): \(\frac{5\pi}{3}\text{ m}\), Length of \(\overset{\frown}{DE}\): \(\frac{5\pi}{2}\text{ m}\), Length of \(\overset{\frown}{GH}\): \(\frac{5\pi}{6}\text{ m}\)
(b) \(\frac{\text{Length of }\overset{\frown}{AB}}{\text{Radius of circle }C}=\frac{5\pi}{6}\), \(\frac{\text{Length of }\overset{\frown}{DE}}{\text{Radius of circle }F}=\frac{5\pi}{6}\), \(\frac{\text{Length of }\overset{\frown}{GH}}{\text{Radius of circle }J}=\frac{5\pi}{6}\)
(c) The length of the arc intercepted by a central angle is proportional to the radius.