Question

in circle a, \\(\angle bae \cong \angle dae\\)

what is the length of \\(\overline{be}\\)?

14 units
17 units
27 units
34 units

Explanation:

<pre_analysis>
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"concepts_used": [
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"Circle Geometry"
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"Congruent Chords Theorem"
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<reasoning>

Equate the chord lengths

\[

$$\begin{aligned} \angle BAE &\cong \angle DAE \implies \widehat{BE} \cong \widehat{DE} \implies BE = DE \\ 3x - 24 &= x + 10 \end{aligned}$$

\]

Solve for x

\[

$$\begin{aligned} 2x &= 34 \\ x &= 17 \end{aligned}$$

\]

Calculate the length of BE

\[

$$\begin{aligned} BE &= 3(17) - 24 \\ BE &= 51 - 24 \\ BE &= 27 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>14 units</mcq-option>
<mcq-option>17 units</mcq-option>
<mcq-correct>27 units</mcq-correct>
<mcq-option>34 units</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circle Geometry"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
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],
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"Circle Geometry"
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"Circle Geometry",
"Congruent Chords Theorem"
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</pre_analysis>

<reasoning>

Equate the chord lengths

\[

$$\begin{aligned} \angle BAE &\cong \angle DAE \implies \widehat{BE} \cong \widehat{DE} \implies BE = DE \\ 3x - 24 &= x + 10 \end{aligned}$$

\]

Solve for x

\[

$$\begin{aligned} 2x &= 34 \\ x &= 17 \end{aligned}$$

\]

Calculate the length of BE

\[

$$\begin{aligned} BE &= 3(17) - 24 \\ BE &= 51 - 24 \\ BE &= 27 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>14 units</mcq-option>
<mcq-option>17 units</mcq-option>
<mcq-correct>27 units</mcq-correct>
<mcq-option>34 units</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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"Circle Geometry"
]
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