Question

1. given the circle below with chords \\(\overline{tu}\\) and \\(\overline{vw}\\). find the length of \\(\overline{wx}\\). round to the nearest tenth if necessary.

Explanation:

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"blank_filling_formatter",
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"new_concepts": [
"Intersecting Chords Theorem"
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"current_concepts": [
"Intersecting Chords Theorem",
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]
}
</pre_analysis>

<reasoning>

Set up the Intersecting Chords Theorem equation

\[
TX \cdot XU = WX \cdot XV
\]

Substitute the given values and solve for WX

\[

$$\begin{aligned} 25 \cdot 39 &= WX \cdot 37 \\ 975 &= 37 \cdot WX \\ WX &= \frac{975}{37} \approx 26.351 \end{aligned}$$

\]

Round to the nearest tenth

\[
WX \approx 26.4
\]
</reasoning>

<answer>
Given the circle below with chords \(\overline{TU}\) and \(\overline{VW}\). Find the length of \(\overline{WX}\). Round to the nearest tenth if necessary.

The length of \(\overline{WX}\) is <blank>26.4</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Intersecting Chords Theorem"
]
}
</post_analysis>

Answer:

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

<reasoning>

Set up the Intersecting Chords Theorem equation

\[
TX \cdot XU = WX \cdot XV
\]

Substitute the given values and solve for WX

\[

$$\begin{aligned} 25 \cdot 39 &= WX \cdot 37 \\ 975 &= 37 \cdot WX \\ WX &= \frac{975}{37} \approx 26.351 \end{aligned}$$

\]

Round to the nearest tenth

\[
WX \approx 26.4
\]
</reasoning>

<answer>
Given the circle below with chords \(\overline{TU}\) and \(\overline{VW}\). Find the length of \(\overline{WX}\). Round to the nearest tenth if necessary.

The length of \(\overline{WX}\) is <blank>26.4</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Intersecting Chords Theorem"
]
}
</post_analysis>