Question

consider circle c with angle acb measuring \\(\frac{3\pi}{4}\\) radians. if minor arc ab measures \\(9\pi\\) inches, what is the length of the radius of circle c? if necessary, round your answer to the nearest inch.

6 inches
12 inches
18 inches
24 inches

Explanation:

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"concepts_used": [
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<reasoning>

State the arc length formula

\[
s = r\theta
\]

Substitute the given values

\[
9\pi = r \cdot \frac{3\pi}{4}
\]

Solve for the radius

\[
r = 9\pi \cdot \frac{4}{3\pi} = 12
\]
</reasoning>

<answer>
<mcq-option>6 inches</mcq-option>
<mcq-correct>12 inches</mcq-correct>
<mcq-option>18 inches</mcq-option>
<mcq-option>24 inches</mcq-option>
</answer>

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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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]
}
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Answer:

<pre_analysis>
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],
"new_concepts": [],
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</pre_analysis>

<reasoning>

State the arc length formula

\[
s = r\theta
\]

Substitute the given values

\[
9\pi = r \cdot \frac{3\pi}{4}
\]

Solve for the radius

\[
r = 9\pi \cdot \frac{4}{3\pi} = 12
\]
</reasoning>

<answer>
<mcq-option>6 inches</mcq-option>
<mcq-correct>12 inches</mcq-correct>
<mcq-option>18 inches</mcq-option>
<mcq-option>24 inches</mcq-option>
</answer>

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"knowledge_point": [
"Mathematics",
"Geometry",
"Arc Length Formula"
]
}
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