Question

find the length of arc abc.
use 3.14 for the value of \\(\pi\\).

? cm
round to the nearest hundredth of a cm.

Explanation:

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<reasoning>

Identify the given parameters

Using the Arc Measure and Central Angle Theorem knowledge points
\[

$$\begin{aligned} &r = 18\text{ cm}\\ &\theta = 120^\circ\\ &\pi \approx 3.14 \end{aligned}$$

\]

Calculate the arc length

Using the Arc Length Formula knowledge point
To find the length of arc \(ABC\), we calculate the fraction of the total circumference corresponding to the central angle:
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]
Substituting the known values:
\[
\text{Arc Length} = \frac{120^\circ}{360^\circ} \times 2 \times 3.14 \times 18
\]
Simplifying the expression:
\[
\text{Arc Length} = \frac{1}{3} \times 37.68 \times 18
\]
\[
\text{Arc Length} = 12 \times 3.14 = 37.68\text{ cm}
\]
</reasoning>

<answer>
Find the length of arc ABC. Use 3.14 for the value of \(\pi\).

<blank>37.68</blank> cm

Round to the nearest hundredth of a cm.
</answer>

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"Mathematics",
"Geometry",
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Answer:

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"Arc Length Formula"
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</pre_analysis>

<reasoning>

Identify the given parameters

Using the Arc Measure and Central Angle Theorem knowledge points
\[

$$\begin{aligned} &r = 18\text{ cm}\\ &\theta = 120^\circ\\ &\pi \approx 3.14 \end{aligned}$$

\]

Calculate the arc length

Using the Arc Length Formula knowledge point
To find the length of arc \(ABC\), we calculate the fraction of the total circumference corresponding to the central angle:
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]
Substituting the known values:
\[
\text{Arc Length} = \frac{120^\circ}{360^\circ} \times 2 \times 3.14 \times 18
\]
Simplifying the expression:
\[
\text{Arc Length} = \frac{1}{3} \times 37.68 \times 18
\]
\[
\text{Arc Length} = 12 \times 3.14 = 37.68\text{ cm}
\]
</reasoning>

<answer>
Find the length of arc ABC. Use 3.14 for the value of \(\pi\).

<blank>37.68</blank> cm

Round to the nearest hundredth of a cm.
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Arc Length Formula"
]
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