below are circles c, f, and j.
a central angle of 135° 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 $ m
length of $ \overparen { d e } $ : $ \square $ m
length of $ \overparen { g h } $ : $ \square $ 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?
the ratios increase by $ \frac { 3 \pi } { 4 } $ units as the radii increase by one unit.
the length of the arc intercepted by a central angle is $ \frac { 3 \pi } { 4 } $ units more than the radius.
the ratios increase by one unit as the lengths of the intercepted arcs increase by $ \frac { 3 \pi } { 4 } $ units.
the length of the arc intercepted by a central angle is proportional to the radius.
none of these.

Question

Accepted Answer

# 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 $\theta = 135^{\circ}$.
## Step2: Calculate the arc - length for circle $C$ ($r = 3m$)
Substitute $r = 3$ and $\theta=135$ into the formula:
$$
\begin{align*}
s_{AB}&=\frac{135}{360}\times2\pi\times3\
&=\frac{3}{8}\times6\pi\
&=\frac{9\pi}{4}\text{ m}
\end{align*}
$$
## Step3: Calculate the arc - length for circle $F$ ($r = 1m$)
Substitute $r = 1$ and $\theta = 135$ into the formula:
$$
\begin{align*}
s_{DE}&=\frac{135}{360}\times2\pi\times1\
&=\frac{3}{8}\times2\pi\
&=\frac{3\pi}{4}\text{ m}
\end{align*}
$$
## Step4: Calculate the arc - length for circle $J$ ($r = 2m$)
Substitute $r = 2$ and $\theta=135$ into the formula:
$$
\begin{align*}
s_{GH}&=\frac{135}{360}\times2\pi\times2\
&=\frac{3}{8}\times4\pi\
&=\frac{3\pi}{2}\text{ m}
\end{align*}
$$
## Step5: Calculate the ratio $\frac{s}{r}$ for circle $C$
$\frac{s_{AB}}{r_C}=\frac{\frac{9\pi}{4}}{3}=\frac{3\pi}{4}$
## Step6: Calculate the ratio $\frac{s}{r}$ for circle $F$
$\frac{s_{DE}}{r_F}=\frac{\frac{3\pi}{4}}{1}=\frac{3\pi}{4}$
## Step7: Calculate the ratio $\frac{s}{r}$ for circle $J$
$\frac{s_{GH}}{r_J}=\frac{\frac{3\pi}{2}}{2}=\frac{3\pi}{4}$

# Answer:
(a) Length of $\overset{\frown}{AB}:\frac{9\pi}{4}\text{ m}$, Length of $\overset{\frown}{DE}:\frac{3\pi}{4}\text{ m}$, Length of $\overset{\frown}{GH}:\frac{3\pi}{2}\text{ m}$
(b) $\frac{\text{Length of }\overset{\frown}{AB}}{\text{Radius of circle }C}=\frac{3\pi}{4}$, $\frac{\text{Length of }\overset{\frown}{DE}}{\text{Radius of circle }F}=\frac{3\pi}{4}$, $\frac{\text{Length of }\overset{\frown}{GH}}{\text{Radius of circle }J}=\frac{3\pi}{4}$
(c) The length of the arc intercepted by a central angle is proportional to the radius.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the arc - length formula

Step2: Calculate the arc - length for circle \(C\) (\(r = 3m\))

Step3: Calculate the arc - length for circle \(F\) (\(r = 1m\))

Step4: Calculate the arc - length for circle \(J\) (\(r = 2m\))

Step5: Calculate the ratio \(\frac{s}{r}\) for circle \(C\)

Step6: Calculate the ratio \(\frac{s}{r}\) for circle \(F\)

Step7: Calculate the ratio \(\frac{s}{r}\) for circle \(J\)

Answer: