Question

circle a has two chords that are equidistant from the center. if \\(de = 6\\), what is the length of \\(cb\\)?

Explanation:

<pre_analysis>
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"quality": "clear",
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"skills_matched": [
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"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Equidistant Chords Theorem"
],
"current_concepts": [
"Equidistant Chords Theorem",
"Circle Geometry"
]
}
</pre_analysis>

<reasoning>

State the Equidistant Chords Theorem

\[
\text{If two chords in a circle are equidistant from the center, then they are congruent in length.}
\]

Apply the theorem to the given circle

\[

$$\begin{aligned} &d(A, CB) = d(A, DE) \implies CB \cong DE \\ &CB = DE \end{aligned}$$

\]

Calculate the length of CB

\[

$$\begin{aligned} &DE = 6 \\ &CB = 6 \end{aligned}$$

\]
</reasoning>

<answer>
6
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Equidistant Chords Theorem"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Equidistant Chords Theorem"
],
"current_concepts": [
"Equidistant Chords Theorem",
"Circle Geometry"
]
}
</pre_analysis>

<reasoning>

State the Equidistant Chords Theorem

\[
\text{If two chords in a circle are equidistant from the center, then they are congruent in length.}
\]

Apply the theorem to the given circle

\[

$$\begin{aligned} &d(A, CB) = d(A, DE) \implies CB \cong DE \\ &CB = DE \end{aligned}$$

\]

Calculate the length of CB

\[

$$\begin{aligned} &DE = 6 \\ &CB = 6 \end{aligned}$$

\]
</reasoning>

<answer>
6
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Equidistant Chords Theorem"
]
}
</post_analysis>